Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.22 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 1558 2835521 2835282 2026-10-02T15:26:02Z ~2026-53001-61 3112001 I added the phrase "What does this mean???" 2835521 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} What <quiz> 1111111-000000=1111111 90+10=100 </quiz> final work <quiz> 10-2=8 212+8+220 scotland forever A A G AG A KAGA KG:AS LAG SGSA"ASG G GSA AGSA GS AG GA GS GSAG AS <nowiki><quiz></nowiki> 22-1=21 23+21(11)=354 <======= What does this mean??? j7yvs4q1u1r05zrrtekbgn4fykdmrkp 5 4327 2835542 2777007 2026-10-02T17:40:30Z Chongwerd 3112008 2835542 wikitext text/x-wiki {{Talkheader}} [[w:Wikipedia:Introduction|Original source of this page at Wikipedia]]. [[Wikiversity:Introduction/archive|some older comments from the test page]] == suggested sequence of tutoria6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!l pages == It is that the sequence of tutorial pages should be:<BR>page 1, "Explore" (state what Wikiversity is and how to find existing content) --> page 2, "edit this page" --> page 3, how to create a new page<BR>--[[User:JWSchmidt|JWSchmidt]] 12:33, 1 September 2006 (UTC) ---- == Differences == Can any one that is there any guideline-article here related to the differece between Wikipedia and Wikiversity. If not it's better to add a guideline as '''Difference between Wikipedia and Wikiversity'''. - [[User:Vaikunda Raja|Vaikunda Raja]] 21:13, 25 November 2006 (UTC) *Wikipedia is for encyclopedia articles. Wikiversity is not for encyclopedia articles. Wikiversity has other types of [[learning resource]]s. There are several Wikiversity pages about Wikipedia. You might start at [[Wikiversity and Wikipedia services]]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) == Science and Psychology Hypothesis == Well lets say that a person has schizophrenia and they have delusions and visions or in their terms hallucinations. Now how much as been done to connect the delusions with a group of schizophrenias. When they hear something it is picked up on brain scans and from my observations using Natural Environmental Studies They seem to produce the same experiences and are and seemingly more inept to esp. How much of this has been observed I am Not for sure. So if any one has any concrete or a theory please let me no the supporting facts thank you *Are you asking to what extent it is possible to find patterns of brain activity that correlate with the experience of hallucinations? Take a look at "[http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=161640 Cerebral activity associated with auditory verbal hallucinations: a functional magnetic resonance imaging case study]" and [http://www.ncbi.nlm.nih.gov/sites/entrez?Db=PubMed&Cmd=ShowDetailView&TermToSearch=17123676&ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstractPlus Neural correlates of inner speech and auditory verbal hallucinations: a critical review and theoretical integration]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) ==Is this page hostile to newcomers?== No doubt someone thought it was cute to build a template into the "leave this line alone" category so this page could have all the lovely html coloring and formatting. Unfortunately it obscures what is going on for newcomers. It might be better to use a plain jane page with obvious wiki syntax that is <b>visible</b> to the newcomers so they can see they are actually editing the page and it is displaying what they told it to edit along with the sections above they were requested to leave alone. I suspect this page confuses rather than enlightens newcomers and thus probably scares a few away who are already timid of programming and markup languages. Time enough for them to learn about templates and leaving them alone on complex pages when they need to edit a main page or a portal page. This should be covered in advanced editing, not the first introductory page a newcomer is asked to edit. [[User:Mirwin|Mirwin]] 07:39, 31 December 2007 (UTC) :"The line above is a template which substitutes in the text the community wishes to remain static or permanent on the top of this introductory page." <-- I'm not sure that this sentence will mean much to new visitors. As someone who has edited for five years, I had to read it three times before I could figure it out. I'm open to experimenting with new ways to encourage people to make their first edit. --[[User:JWS|JWS]] 15:12, 31 December 2007 (UTC) == Interlingual == This resource is great. I have imported it to beta. Does anybody (or Erkan, or Juan?) know how to find similar pages in other languages? --[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 05:27, 20 September 2008 (UTC) --[[User:Ddreview|Ddreview]] 02:16, 26 November 2008 (UTC)== 112508.ddreview - something basic == suffered a case of situational depression. Slowly I continue to recover and even slower at realizing a measurable return in faculties. Cognitive recall and memory - enough about explaining... I am a professional land surveyor with a specific focus on inland and riparian boundary law. I am submitting a narrative with respect to my capabilities and worth to an energy business. I'm challenged in remembering or knowing the meaning of each and relative to each other and which is correct to use where and when. knowledge, intellect, experience, understanding, intelligence, expertise, acumen, and wisdom thank you for your patience 0=0 0*1=0*1 0/0*1=1 0/0=0 == 0/0=1 == 0=0 0*1=0*1 0/0*1=1 0/0=1 The one who wrote the previous equation should understand that mathematics never plays such jokes. First of all dividing by zero is not accepted in Mathematics unless the value is infinity. See the [http://en.wikipedia.org/wiki/Rational_number | Rational number] definition for more understanding. Finally 0/0 is an illegal term in Maths regardless the rest of the mathematical expression, and that's why limits are used. For more information about these tricks visit this wikipedia page [http://en.wikipedia.org/wiki/Division_by_zero#Fallacies_based_on_division_by_zero | Fallacies based on division by zero]--[[User:Email4mobile|Email4mobile]] 02:51, 5 June 2009 (UTC) == here I boldly edit == Is there a Wiki-Kidversity ??? ( Kids are generally alot smarter than most teachers realize, and if a simplified interface was made, kids could start to contribute an entire universe of content. ) One potential "Wiki-Growth Exponentiator" might be finding a way to better integrate online social networking to bring in a new breed of cyber netizen nerds. The people of www.szstuff.com are a good example of people who seek to teach and learn. Online social networking has an addictive element... The most effective learning is that which fully engages an array of one's senses. Uni students love listening to and making music, right? Can't wikiversity have featured pages for wikibands to share musical knowledge whilst collaborating on other academic projects? That's my two cents. Spend it wisely. ;) {{unsigned|andrew.aker|04:22, 30 September 2009}} : Some good ideas with potential here. For some possible starting points, try: <div style="float:right; width:100%"> {{gateways}} </div> -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:00, 30 September 2009 (UTC) == bold editing not taken kindly to in these parts == gimme a break for editing the wrong way. it was my first edit. anyway, i thought it fit better where it was than the comment that currently resides there. who says they can override my comment? not cool wikiversidad == Protect this page? == I've just reverted page blanking practically twice in a row over here. It looks like too many people, intentionally or otherwise, are destroying the beautiful content we've put on this page. Perhaps the page needs a bit more protection than it's been given. For people who want to make their first edit, why don't we direct them to a blank page that we have open in a new window? --[[User:AFriedman|AFriedman]] 07:42, 25 October 2009 (UTC) : We could modify the page to send people to [[Sandbox]] for editing (which has a bot to tidy it) - or an empty-ish subpage? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:27, 28 October 2009 (UTC) :: I think the intention was to send them to a sandbox page, for edits but navigating to a sandbox page is too advanced for most starting editors. Perhaps link the Edit button on the page to a sandbox page with a translocation of an explanation of what is different here.--[[User:Graeme E. Smith|Graeme E. Smith]] 14:30, 28 October 2009 (UTC) :::Graeme, I like that idea. Or, protect the page and give readers a link to the sandbox (which would open in a new window) so readers don't change the original page? Basically, something that is workable for new users and would stop the periodic destruction to the page. --[[User:AFriedman|AFriedman]] 02:38, 29 October 2009 (UTC) :::: See what you think about [http://en.wikiversity.org/w/index.php?title=Template%3APlease_leave_this_line_alone&action=historysubmit&diff=497324&oldid=433089 these changes]. I think we encourage [[sandbox]] editing and then there is one less place to monitor. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:46, 29 October 2009 (UTC) Is there any way to have the link to the sandbox open in a new window or tab? Otherwise, I am concerned that going to the sandbox might get new users lost. --[[User:AFriedman|AFriedman]] 18:47, 29 October 2009 (UTC) : Had a quick play/search but couldn't get anything working here from [[mw:Manual:Opening external links in a new window]]. Whilst it may be short-term helpful, since this would not be normal navigation (opening a new window), it could actually be more confusing. So, at the moment I'm ambivalent about putting in non-normal navigation (opening a link in a new window/tab). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:59, 29 October 2009 (UTC) ::Thanks for sharing the MediaWiki page. I was looking at the page and was wondering how you changed it to show the link to the sandbox? Also, I accidentally blanked the page myself while I was experimenting. It seems like this would be awfully easy to do even for users in good faith, and hard to revert for people who don't know what they're doing. Do you think anything is lost if we protect the page? Before you made the link to the sandbox, non-experimental edits hadn't happened there in a long time. --[[User:AFriedman|AFriedman]] 01:03, 30 October 2009 (UTC) == Video "Wikiversity Special Report" learn the wiki way == I tried multiple versions of the file, only to find that most of them were garbled in one way or another. Most gave me a greyscreen, with shadows, started out very choppy, and I didn't recognize that they were playing music in the intro. The best version was QuickTime, but even it, made the womans voice virtually indecipherable. The main text was good however--[[User:Graeme E. Smith|Graeme E. Smith]] 16:12, 1 January 2010 (UTC) == Romanian Wikiversitate == Could anyone create a Romanian Wikiversitate portal? Thank you. [[User:Calusarul|Calusarul]] 22:18, 17 February 2010 (UTC) :Please see at [http://beta.wikiversity.org/wiki/Pagina_principal%C4%83 beta wikiversity], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduce myself == '''My name is Kanadee, this name is a unique name in Thai language because it's a bit strange. it means 'Now Good' in Thai. However you can call me as my nickname 'Poy'. Now im 2nd year in MUIC. i attend many activities in my college like welcoming camp, as im p'group, and volunteer club. i love fashion n beauty. i always make my friends to get better styles. Hope tht now u will know me better.''' ----------------------------------------------------------------------------------------------------------------------------------------------------- '''My name is Dr. Bitetto. I am currently a science researcher in the field of robotics and control systems. My education starts with my Bachelor's of Science in Computer Systems Engineering from the State University of New York (1991) and continues through my Doctoral degree in Robotics from the Union Institute (1994). I have taught both at the graduate level in a university and also at the professional level at an adult continuing educational service. ''' '''I currently hold numerous patents and copyrights from both my undergraduate and doctoral level work. Presently, I am employed by the Institute of Cybernetics Research, Inc. (see my linkedin web-page for Dr. Marco Bitetto), where I am engaged as a science researcher doing research in the areas of feasibility analysis of designs for robots and control systems.''' --[[User:Robodoc|Robodoc]] ([[User talk:Robodoc|talk]]) 13:42, 20 August 2012 (UTC) ----------------------------------------------------------------------------------------------------------------------------------------------------- My name is Aviwe I'm still in high school but I have special subject like cat , business studies,economics and mathematics lit ..but I want to be sport management [[User:Aviwe memeza|Aviwe memeza]] ([[User talk:Aviwe memeza|discuss]] • [[Special:Contributions/Aviwe memeza|contribs]]) 07:56, 25 March 2021 (UTC) == bluetooth connectivity == which type of network or signals are being used in bluetooth? how bluetooth works? :Please use [[Wikiversity:Help desk]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduction == Is it me, or is the intro thing entirely based around introducing the concept of a wiki rather than introducing specifically wikiversity? Because I'm kind of finding the intro and front page to be somewhat vague on the subject of what exactly wikiversity is and does. It doesn't really lend itself to immediate comprehension of how the community of "learning resources" works or how to jump in to one of those learning communities. Perhaps this could be elucidated? [[Special:Contributions/99.23.240.60|99.23.240.60]] 14:19, 9 May 2012 (UTC) == can any one tell the applications of harmonographs?? == Harmonographs are the complex curves drawn with the help of two or more pendulum. Is their any proper application and use of those curves in general life expect for drawing complex curves for fun???--[[User:Gyandeeprkmv|Gyandeeprkmv]] ([[User talk:Gyandeeprkmv|talk]]) 19:03, 5 August 2012 (UTC) == Image sizes == Is it possible that there may be something wrong with this page? I am getting huge images on my screen and there is no text. something is out of whack. I need to read this before I fix anything :D [[User:Svrangerchrista|Svrangerchrista]] ([[User talk:Svrangerchrista|discuss]] • [[Special:Contributions/Svrangerchrista|contribs]]) 05:02, 22 April 2017 (UTC) :Now corrected. Thanks for letting us know. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:47, 22 April 2017 (UTC) == Phần mở đầu == Mở đầu [[User:XuanBach226700|XuanBach226700]] ([[User talk:XuanBach226700|discuss]] • [[Special:Contributions/XuanBach226700|contribs]]) 13:51, 13 May 2020 (UTC) == Climate change == What is climate change? [[User:Zonkerh|Zonkerh]] ([[User talk:Zonkerh|discuss]] • [[Special:Contributions/Zonkerh|contribs]]) 18:30, 26 June 2020 (UTC) :{{At|Zonkerh}} See [[Wikipedia: Global warming]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:25, 26 June 2020 (UTC) oyafznfozow41wypabnprav6q5h8flr 2835543 2835542 2026-10-02T17:41:11Z Chongwerd 3112008 2835543 wikitext text/x-wiki {{Talkheader}} [[w:Wikipedia:Introduction|Original source of this page at Wikipedia]]. [[Wikiversity:Introduction/archive|some older comments from the test page]] == suggested sequence of tutoria6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!l pages == It is that the sequence of tutorial pages should be:<BR>page 1,6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!!6767676767767 drizzzy AHHHHH!!!!!! "Explore" (state what Wikiversity is and how to find existing content) --> page 2, "edit this page" --> page 3, how to create a new page<BR>--[[User:JWSchmidt|JWSchmidt]] 12:33, 1 September 2006 (UTC) ---- == Differences == Can any one that is there any guideline-article here related to the differece between Wikipedia and Wikiversity. If not it's better to add a guideline as '''Difference between Wikipedia and Wikiversity'''. - [[User:Vaikunda Raja|Vaikunda Raja]] 21:13, 25 November 2006 (UTC) *Wikipedia is for encyclopedia articles. Wikiversity is not for encyclopedia articles. Wikiversity has other types of [[learning resource]]s. There are several Wikiversity pages about Wikipedia. You might start at [[Wikiversity and Wikipedia services]]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) == Science and Psychology Hypothesis == Well lets say that a person has schizophrenia and they have delusions and visions or in their terms hallucinations. Now how much as been done to connect the delusions with a group of schizophrenias. When they hear something it is picked up on brain scans and from my observations using Natural Environmental Studies They seem to produce the same experiences and are and seemingly more inept to esp. How much of this has been observed I am Not for sure. So if any one has any concrete or a theory please let me no the supporting facts thank you *Are you asking to what extent it is possible to find patterns of brain activity that correlate with the experience of hallucinations? Take a look at "[http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=161640 Cerebral activity associated with auditory verbal hallucinations: a functional magnetic resonance imaging case study]" and [http://www.ncbi.nlm.nih.gov/sites/entrez?Db=PubMed&Cmd=ShowDetailView&TermToSearch=17123676&ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstractPlus Neural correlates of inner speech and auditory verbal hallucinations: a critical review and theoretical integration]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) ==Is this page hostile to newcomers?== No doubt someone thought it was cute to build a template into the "leave this line alone" category so this page could have all the lovely html coloring and formatting. Unfortunately it obscures what is going on for newcomers. It might be better to use a plain jane page with obvious wiki syntax that is <b>visible</b> to the newcomers so they can see they are actually editing the page and it is displaying what they told it to edit along with the sections above they were requested to leave alone. I suspect this page confuses rather than enlightens newcomers and thus probably scares a few away who are already timid of programming and markup languages. Time enough for them to learn about templates and leaving them alone on complex pages when they need to edit a main page or a portal page. This should be covered in advanced editing, not the first introductory page a newcomer is asked to edit. [[User:Mirwin|Mirwin]] 07:39, 31 December 2007 (UTC) :"The line above is a template which substitutes in the text the community wishes to remain static or permanent on the top of this introductory page." <-- I'm not sure that this sentence will mean much to new visitors. As someone who has edited for five years, I had to read it three times before I could figure it out. I'm open to experimenting with new ways to encourage people to make their first edit. --[[User:JWS|JWS]] 15:12, 31 December 2007 (UTC) == Interlingual == This resource is great. I have imported it to beta. Does anybody (or Erkan, or Juan?) know how to find similar pages in other languages? --[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 05:27, 20 September 2008 (UTC) --[[User:Ddreview|Ddreview]] 02:16, 26 November 2008 (UTC)== 112508.ddreview - something basic == suffered a case of situational depression. Slowly I continue to recover and even slower at realizing a measurable return in faculties. Cognitive recall and memory - enough about explaining... I am a professional land surveyor with a specific focus on inland and riparian boundary law. I am submitting a narrative with respect to my capabilities and worth to an energy business. I'm challenged in remembering or knowing the meaning of each and relative to each other and which is correct to use where and when. knowledge, intellect, experience, understanding, intelligence, expertise, acumen, and wisdom thank you for your patience 0=0 0*1=0*1 0/0*1=1 0/0=0 == 0/0=1 == 0=0 0*1=0*1 0/0*1=1 0/0=1 The one who wrote the previous equation should understand that mathematics never plays such jokes. First of all dividing by zero is not accepted in Mathematics unless the value is infinity. See the [http://en.wikipedia.org/wiki/Rational_number | Rational number] definition for more understanding. Finally 0/0 is an illegal term in Maths regardless the rest of the mathematical expression, and that's why limits are used. For more information about these tricks visit this wikipedia page [http://en.wikipedia.org/wiki/Division_by_zero#Fallacies_based_on_division_by_zero | Fallacies based on division by zero]--[[User:Email4mobile|Email4mobile]] 02:51, 5 June 2009 (UTC) == here I boldly edit == Is there a Wiki-Kidversity ??? ( Kids are generally alot smarter than most teachers realize, and if a simplified interface was made, kids could start to contribute an entire universe of content. ) One potential "Wiki-Growth Exponentiator" might be finding a way to better integrate online social networking to bring in a new breed of cyber netizen nerds. The people of www.szstuff.com are a good example of people who seek to teach and learn. Online social networking has an addictive element... The most effective learning is that which fully engages an array of one's senses. Uni students love listening to and making music, right? Can't wikiversity have featured pages for wikibands to share musical knowledge whilst collaborating on other academic projects? That's my two cents. Spend it wisely. ;) {{unsigned|andrew.aker|04:22, 30 September 2009}} : Some good ideas with potential here. For some possible starting points, try: <div style="float:right; width:100%"> {{gateways}} </div> -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:00, 30 September 2009 (UTC) == bold editing not taken kindly to in these parts == gimme a break for editing the wrong way. it was my first edit. anyway, i thought it fit better where it was than the comment that currently resides there. who says they can override my comment? not cool wikiversidad == Protect this page? == I've just reverted page blanking practically twice in a row over here. It looks like too many people, intentionally or otherwise, are destroying the beautiful content we've put on this page. Perhaps the page needs a bit more protection than it's been given. For people who want to make their first edit, why don't we direct them to a blank page that we have open in a new window? --[[User:AFriedman|AFriedman]] 07:42, 25 October 2009 (UTC) : We could modify the page to send people to [[Sandbox]] for editing (which has a bot to tidy it) - or an empty-ish subpage? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:27, 28 October 2009 (UTC) :: I think the intention was to send them to a sandbox page, for edits but navigating to a sandbox page is too advanced for most starting editors. Perhaps link the Edit button on the page to a sandbox page with a translocation of an explanation of what is different here.--[[User:Graeme E. Smith|Graeme E. Smith]] 14:30, 28 October 2009 (UTC) :::Graeme, I like that idea. Or, protect the page and give readers a link to the sandbox (which would open in a new window) so readers don't change the original page? Basically, something that is workable for new users and would stop the periodic destruction to the page. --[[User:AFriedman|AFriedman]] 02:38, 29 October 2009 (UTC) :::: See what you think about [http://en.wikiversity.org/w/index.php?title=Template%3APlease_leave_this_line_alone&action=historysubmit&diff=497324&oldid=433089 these changes]. I think we encourage [[sandbox]] editing and then there is one less place to monitor. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:46, 29 October 2009 (UTC) Is there any way to have the link to the sandbox open in a new window or tab? Otherwise, I am concerned that going to the sandbox might get new users lost. --[[User:AFriedman|AFriedman]] 18:47, 29 October 2009 (UTC) : Had a quick play/search but couldn't get anything working here from [[mw:Manual:Opening external links in a new window]]. Whilst it may be short-term helpful, since this would not be normal navigation (opening a new window), it could actually be more confusing. So, at the moment I'm ambivalent about putting in non-normal navigation (opening a link in a new window/tab). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:59, 29 October 2009 (UTC) ::Thanks for sharing the MediaWiki page. I was looking at the page and was wondering how you changed it to show the link to the sandbox? Also, I accidentally blanked the page myself while I was experimenting. It seems like this would be awfully easy to do even for users in good faith, and hard to revert for people who don't know what they're doing. Do you think anything is lost if we protect the page? Before you made the link to the sandbox, non-experimental edits hadn't happened there in a long time. --[[User:AFriedman|AFriedman]] 01:03, 30 October 2009 (UTC) == Video "Wikiversity Special Report" learn the wiki way == I tried multiple versions of the file, only to find that most of them were garbled in one way or another. Most gave me a greyscreen, with shadows, started out very choppy, and I didn't recognize that they were playing music in the intro. The best version was QuickTime, but even it, made the womans voice virtually indecipherable. The main text was good however--[[User:Graeme E. Smith|Graeme E. Smith]] 16:12, 1 January 2010 (UTC) == Romanian Wikiversitate == Could anyone create a Romanian Wikiversitate portal? Thank you. [[User:Calusarul|Calusarul]] 22:18, 17 February 2010 (UTC) :Please see at [http://beta.wikiversity.org/wiki/Pagina_principal%C4%83 beta wikiversity], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduce myself == '''My name is Kanadee, this name is a unique name in Thai language because it's a bit strange. it means 'Now Good' in Thai. However you can call me as my nickname 'Poy'. Now im 2nd year in MUIC. i attend many activities in my college like welcoming camp, as im p'group, and volunteer club. i love fashion n beauty. i always make my friends to get better styles. Hope tht now u will know me better.''' ----------------------------------------------------------------------------------------------------------------------------------------------------- '''My name is Dr. Bitetto. I am currently a science researcher in the field of robotics and control systems. My education starts with my Bachelor's of Science in Computer Systems Engineering from the State University of New York (1991) and continues through my Doctoral degree in Robotics from the Union Institute (1994). I have taught both at the graduate level in a university and also at the professional level at an adult continuing educational service. ''' '''I currently hold numerous patents and copyrights from both my undergraduate and doctoral level work. Presently, I am employed by the Institute of Cybernetics Research, Inc. (see my linkedin web-page for Dr. Marco Bitetto), where I am engaged as a science researcher doing research in the areas of feasibility analysis of designs for robots and control systems.''' --[[User:Robodoc|Robodoc]] ([[User talk:Robodoc|talk]]) 13:42, 20 August 2012 (UTC) ----------------------------------------------------------------------------------------------------------------------------------------------------- My name is Aviwe I'm still in high school but I have special subject like cat , business studies,economics and mathematics lit ..but I want to be sport management [[User:Aviwe memeza|Aviwe memeza]] ([[User talk:Aviwe memeza|discuss]] • [[Special:Contributions/Aviwe memeza|contribs]]) 07:56, 25 March 2021 (UTC) == bluetooth connectivity == which type of network or signals are being used in bluetooth? how bluetooth works? :Please use [[Wikiversity:Help desk]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduction == Is it me, or is the intro thing entirely based around introducing the concept of a wiki rather than introducing specifically wikiversity? Because I'm kind of finding the intro and front page to be somewhat vague on the subject of what exactly wikiversity is and does. It doesn't really lend itself to immediate comprehension of how the community of "learning resources" works or how to jump in to one of those learning communities. Perhaps this could be elucidated? [[Special:Contributions/99.23.240.60|99.23.240.60]] 14:19, 9 May 2012 (UTC) == can any one tell the applications of harmonographs?? == Harmonographs are the complex curves drawn with the help of two or more pendulum. Is their any proper application and use of those curves in general life expect for drawing complex curves for fun???--[[User:Gyandeeprkmv|Gyandeeprkmv]] ([[User talk:Gyandeeprkmv|talk]]) 19:03, 5 August 2012 (UTC) == Image sizes == Is it possible that there may be something wrong with this page? I am getting huge images on my screen and there is no text. something is out of whack. I need to read this before I fix anything :D [[User:Svrangerchrista|Svrangerchrista]] ([[User talk:Svrangerchrista|discuss]] • [[Special:Contributions/Svrangerchrista|contribs]]) 05:02, 22 April 2017 (UTC) :Now corrected. Thanks for letting us know. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:47, 22 April 2017 (UTC) == Phần mở đầu == Mở đầu [[User:XuanBach226700|XuanBach226700]] ([[User talk:XuanBach226700|discuss]] • [[Special:Contributions/XuanBach226700|contribs]]) 13:51, 13 May 2020 (UTC) == Climate change == What is climate change? [[User:Zonkerh|Zonkerh]] ([[User talk:Zonkerh|discuss]] • [[Special:Contributions/Zonkerh|contribs]]) 18:30, 26 June 2020 (UTC) :{{At|Zonkerh}} See [[Wikipedia: Global warming]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:25, 26 June 2020 (UTC) bs1ckj0xm8lfso4qx2tyy5tg6dw20g3 2835545 2835543 2026-10-02T17:43:43Z NDG 3006541 Reverted edits by [[Special:Contributions/Chongwerd|Chongwerd]] ([[User_talk:Chongwerd|talk]]) to last version by [[User:Atcovi|Atcovi]] using [[Wikiversity:Rollback|rollback]] 2612322 wikitext text/x-wiki {{Talkheader}} [[w:Wikipedia:Introduction|Original source of this page at Wikipedia]]. [[Wikiversity:Introduction/archive|some older comments from the test page]] == suggested sequence of tutorial pages == It is that the sequence of tutorial pages should be:<BR>page 1, "Explore" (state what Wikiversity is and how to find existing content) --> page 2, "edit this page" --> page 3, how to create a new page<BR>--[[User:JWSchmidt|JWSchmidt]] 12:33, 1 September 2006 (UTC) ---- == Differences == Can any one that is there any guideline-article here related to the differece between Wikipedia and Wikiversity. If not it's better to add a guideline as '''Difference between Wikipedia and Wikiversity'''. - [[User:Vaikunda Raja|Vaikunda Raja]] 21:13, 25 November 2006 (UTC) *Wikipedia is for encyclopedia articles. Wikiversity is not for encyclopedia articles. Wikiversity has other types of [[learning resource]]s. There are several Wikiversity pages about Wikipedia. You might start at [[Wikiversity and Wikipedia services]]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) == Science and Psychology Hypothesis == Well lets say that a person has schizophrenia and they have delusions and visions or in their terms hallucinations. Now how much as been done to connect the delusions with a group of schizophrenias. When they hear something it is picked up on brain scans and from my observations using Natural Environmental Studies They seem to produce the same experiences and are and seemingly more inept to esp. How much of this has been observed I am Not for sure. So if any one has any concrete or a theory please let me no the supporting facts thank you *Are you asking to what extent it is possible to find patterns of brain activity that correlate with the experience of hallucinations? Take a look at "[http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=161640 Cerebral activity associated with auditory verbal hallucinations: a functional magnetic resonance imaging case study]" and [http://www.ncbi.nlm.nih.gov/sites/entrez?Db=PubMed&Cmd=ShowDetailView&TermToSearch=17123676&ordinalpos=1&itool=EntrezSystem2.PEntrez.Pubmed.Pubmed_ResultsPanel.Pubmed_RVAbstractPlus Neural correlates of inner speech and auditory verbal hallucinations: a critical review and theoretical integration]. --[[User:JWSchmidt|JWSchmidt]] 17:16, 19 July 2007 (UTC) ==Is this page hostile to newcomers?== No doubt someone thought it was cute to build a template into the "leave this line alone" category so this page could have all the lovely html coloring and formatting. Unfortunately it obscures what is going on for newcomers. It might be better to use a plain jane page with obvious wiki syntax that is <b>visible</b> to the newcomers so they can see they are actually editing the page and it is displaying what they told it to edit along with the sections above they were requested to leave alone. I suspect this page confuses rather than enlightens newcomers and thus probably scares a few away who are already timid of programming and markup languages. Time enough for them to learn about templates and leaving them alone on complex pages when they need to edit a main page or a portal page. This should be covered in advanced editing, not the first introductory page a newcomer is asked to edit. [[User:Mirwin|Mirwin]] 07:39, 31 December 2007 (UTC) :"The line above is a template which substitutes in the text the community wishes to remain static or permanent on the top of this introductory page." <-- I'm not sure that this sentence will mean much to new visitors. As someone who has edited for five years, I had to read it three times before I could figure it out. I'm open to experimenting with new ways to encourage people to make their first edit. --[[User:JWS|JWS]] 15:12, 31 December 2007 (UTC) == Interlingual == This resource is great. I have imported it to beta. Does anybody (or Erkan, or Juan?) know how to find similar pages in other languages? --[[User:Hillgentleman|Hillgentleman]]|[[User talk:hillgentleman|Talk]] 05:27, 20 September 2008 (UTC) --[[User:Ddreview|Ddreview]] 02:16, 26 November 2008 (UTC)== 112508.ddreview - something basic == suffered a case of situational depression. Slowly I continue to recover and even slower at realizing a measurable return in faculties. Cognitive recall and memory - enough about explaining... I am a professional land surveyor with a specific focus on inland and riparian boundary law. I am submitting a narrative with respect to my capabilities and worth to an energy business. I'm challenged in remembering or knowing the meaning of each and relative to each other and which is correct to use where and when. knowledge, intellect, experience, understanding, intelligence, expertise, acumen, and wisdom thank you for your patience 0=0 0*1=0*1 0/0*1=1 0/0=0 == 0/0=1 == 0=0 0*1=0*1 0/0*1=1 0/0=1 The one who wrote the previous equation should understand that mathematics never plays such jokes. First of all dividing by zero is not accepted in Mathematics unless the value is infinity. See the [http://en.wikipedia.org/wiki/Rational_number | Rational number] definition for more understanding. Finally 0/0 is an illegal term in Maths regardless the rest of the mathematical expression, and that's why limits are used. For more information about these tricks visit this wikipedia page [http://en.wikipedia.org/wiki/Division_by_zero#Fallacies_based_on_division_by_zero | Fallacies based on division by zero]--[[User:Email4mobile|Email4mobile]] 02:51, 5 June 2009 (UTC) == here I boldly edit == Is there a Wiki-Kidversity ??? ( Kids are generally alot smarter than most teachers realize, and if a simplified interface was made, kids could start to contribute an entire universe of content. ) One potential "Wiki-Growth Exponentiator" might be finding a way to better integrate online social networking to bring in a new breed of cyber netizen nerds. The people of www.szstuff.com are a good example of people who seek to teach and learn. Online social networking has an addictive element... The most effective learning is that which fully engages an array of one's senses. Uni students love listening to and making music, right? Can't wikiversity have featured pages for wikibands to share musical knowledge whilst collaborating on other academic projects? That's my two cents. Spend it wisely. ;) {{unsigned|andrew.aker|04:22, 30 September 2009}} : Some good ideas with potential here. For some possible starting points, try: <div style="float:right; width:100%"> {{gateways}} </div> -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:00, 30 September 2009 (UTC) == bold editing not taken kindly to in these parts == gimme a break for editing the wrong way. it was my first edit. anyway, i thought it fit better where it was than the comment that currently resides there. who says they can override my comment? not cool wikiversidad == Protect this page? == I've just reverted page blanking practically twice in a row over here. It looks like too many people, intentionally or otherwise, are destroying the beautiful content we've put on this page. Perhaps the page needs a bit more protection than it's been given. For people who want to make their first edit, why don't we direct them to a blank page that we have open in a new window? --[[User:AFriedman|AFriedman]] 07:42, 25 October 2009 (UTC) : We could modify the page to send people to [[Sandbox]] for editing (which has a bot to tidy it) - or an empty-ish subpage? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:27, 28 October 2009 (UTC) :: I think the intention was to send them to a sandbox page, for edits but navigating to a sandbox page is too advanced for most starting editors. Perhaps link the Edit button on the page to a sandbox page with a translocation of an explanation of what is different here.--[[User:Graeme E. Smith|Graeme E. Smith]] 14:30, 28 October 2009 (UTC) :::Graeme, I like that idea. Or, protect the page and give readers a link to the sandbox (which would open in a new window) so readers don't change the original page? Basically, something that is workable for new users and would stop the periodic destruction to the page. --[[User:AFriedman|AFriedman]] 02:38, 29 October 2009 (UTC) :::: See what you think about [http://en.wikiversity.org/w/index.php?title=Template%3APlease_leave_this_line_alone&action=historysubmit&diff=497324&oldid=433089 these changes]. I think we encourage [[sandbox]] editing and then there is one less place to monitor. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:46, 29 October 2009 (UTC) Is there any way to have the link to the sandbox open in a new window or tab? Otherwise, I am concerned that going to the sandbox might get new users lost. --[[User:AFriedman|AFriedman]] 18:47, 29 October 2009 (UTC) : Had a quick play/search but couldn't get anything working here from [[mw:Manual:Opening external links in a new window]]. Whilst it may be short-term helpful, since this would not be normal navigation (opening a new window), it could actually be more confusing. So, at the moment I'm ambivalent about putting in non-normal navigation (opening a link in a new window/tab). -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 21:59, 29 October 2009 (UTC) ::Thanks for sharing the MediaWiki page. I was looking at the page and was wondering how you changed it to show the link to the sandbox? Also, I accidentally blanked the page myself while I was experimenting. It seems like this would be awfully easy to do even for users in good faith, and hard to revert for people who don't know what they're doing. Do you think anything is lost if we protect the page? Before you made the link to the sandbox, non-experimental edits hadn't happened there in a long time. --[[User:AFriedman|AFriedman]] 01:03, 30 October 2009 (UTC) == Video "Wikiversity Special Report" learn the wiki way == I tried multiple versions of the file, only to find that most of them were garbled in one way or another. Most gave me a greyscreen, with shadows, started out very choppy, and I didn't recognize that they were playing music in the intro. The best version was QuickTime, but even it, made the womans voice virtually indecipherable. The main text was good however--[[User:Graeme E. Smith|Graeme E. Smith]] 16:12, 1 January 2010 (UTC) == Romanian Wikiversitate == Could anyone create a Romanian Wikiversitate portal? Thank you. [[User:Calusarul|Calusarul]] 22:18, 17 February 2010 (UTC) :Please see at [http://beta.wikiversity.org/wiki/Pagina_principal%C4%83 beta wikiversity], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduce myself == '''My name is Kanadee, this name is a unique name in Thai language because it's a bit strange. it means 'Now Good' in Thai. However you can call me as my nickname 'Poy'. Now im 2nd year in MUIC. i attend many activities in my college like welcoming camp, as im p'group, and volunteer club. i love fashion n beauty. i always make my friends to get better styles. Hope tht now u will know me better.''' ----------------------------------------------------------------------------------------------------------------------------------------------------- '''My name is Dr. Bitetto. I am currently a science researcher in the field of robotics and control systems. My education starts with my Bachelor's of Science in Computer Systems Engineering from the State University of New York (1991) and continues through my Doctoral degree in Robotics from the Union Institute (1994). I have taught both at the graduate level in a university and also at the professional level at an adult continuing educational service. ''' '''I currently hold numerous patents and copyrights from both my undergraduate and doctoral level work. Presently, I am employed by the Institute of Cybernetics Research, Inc. (see my linkedin web-page for Dr. Marco Bitetto), where I am engaged as a science researcher doing research in the areas of feasibility analysis of designs for robots and control systems.''' --[[User:Robodoc|Robodoc]] ([[User talk:Robodoc|talk]]) 13:42, 20 August 2012 (UTC) ----------------------------------------------------------------------------------------------------------------------------------------------------- My name is Aviwe I'm still in high school but I have special subject like cat , business studies,economics and mathematics lit ..but I want to be sport management [[User:Aviwe memeza|Aviwe memeza]] ([[User talk:Aviwe memeza|discuss]] • [[Special:Contributions/Aviwe memeza|contribs]]) 07:56, 25 March 2021 (UTC) == bluetooth connectivity == which type of network or signals are being used in bluetooth? how bluetooth works? :Please use [[Wikiversity:Help desk]], ----[[User:Erkan_Yilmaz|Erkan Yilmaz]] <small>uses the [[Wikiversity:Chat]]</small> 18:12, 14 January 2011 (UTC) == Introduction == Is it me, or is the intro thing entirely based around introducing the concept of a wiki rather than introducing specifically wikiversity? Because I'm kind of finding the intro and front page to be somewhat vague on the subject of what exactly wikiversity is and does. It doesn't really lend itself to immediate comprehension of how the community of "learning resources" works or how to jump in to one of those learning communities. Perhaps this could be elucidated? [[Special:Contributions/99.23.240.60|99.23.240.60]] 14:19, 9 May 2012 (UTC) == can any one tell the applications of harmonographs?? == Harmonographs are the complex curves drawn with the help of two or more pendulum. Is their any proper application and use of those curves in general life expect for drawing complex curves for fun???--[[User:Gyandeeprkmv|Gyandeeprkmv]] ([[User talk:Gyandeeprkmv|talk]]) 19:03, 5 August 2012 (UTC) == Image sizes == Is it possible that there may be something wrong with this page? I am getting huge images on my screen and there is no text. something is out of whack. I need to read this before I fix anything :D [[User:Svrangerchrista|Svrangerchrista]] ([[User talk:Svrangerchrista|discuss]] • [[Special:Contributions/Svrangerchrista|contribs]]) 05:02, 22 April 2017 (UTC) :Now corrected. Thanks for letting us know. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:47, 22 April 2017 (UTC) == Phần mở đầu == Mở đầu [[User:XuanBach226700|XuanBach226700]] ([[User talk:XuanBach226700|discuss]] • [[Special:Contributions/XuanBach226700|contribs]]) 13:51, 13 May 2020 (UTC) == Climate change == What is climate change? [[User:Zonkerh|Zonkerh]] ([[User talk:Zonkerh|discuss]] • [[Special:Contributions/Zonkerh|contribs]]) 18:30, 26 June 2020 (UTC) :{{At|Zonkerh}} See [[Wikipedia: Global warming]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:25, 26 June 2020 (UTC) d9wklootca6kg15a0nk5dvpuab7cven 0 37372 2835574 2834858 2026-10-02T21:04:11Z Evgenii Arsentev 3110906 Sub-pages: link new learning resource on standing instructions for AI coding agents 2835574 wikitext text/x-wiki {{AI_navbar_0}} {{RightTOC}} Artificial intelligence involves intelligence that is created through computers or other means other than humanness. This area is for learning, teaching, and research related to artificial intelligence. AI usage has evolved into [[agentic harnesses]] and more. ==Learning resources== * [[Artificial neural network]] * [[Cross-domain AI topics]] ===Sub-pages === * [[/Introduction/]] * [[Artificial intelligence/prospects|Prospects for artificial intelligence]] ** [[Supersymmetry/String Theory based Artificial Intelligence]] * [[/Lecture aid/]] * [[/AI content engine economics/]] * [[/Checking AI output with a second agent/]] * [[/Giving an AI agent access safely/]] * [[/Checking an open-source AI project before you depend on it/]] * [[/Standing instructions for an AI coding agent/]] ==Discussion questions and essay ideas== * Can artificial intelligence (AI) be used as a tool for good? List applications scenarios of AI. Discuss ethical aspects that are related to decision making and decision support with AI. Define requirements and constraints of AI application that you could derive from your analysis. * Identify areas of application where AI that can assist staff members e.g. in health care facilities to get more time for social interaction and support for patients? What happens if AI applications fail and patients get harmed? What level of security must be applied and who is responsible for that. Apply the underlying concept on other topics like autonomous driving, ...) * Analyse the concept of [[machine learning]] and analysis of big data! What is the importance (PRO and CONs) of AI in this context in comparision to other mathematical or computer science methodologies? * Assume AI is used for pattern recognition in social media data or in general in the context of [[Commercial Data Harvesting]]. Provide examples and analyse the possible consequence of false-positive resp. false negative classification of individuals! What would be your response and what would you suggest to improve the considered system? * How can the human brain or human mind be emulated using computers? Is it desirable to? * What other ethical issues are involved with artificial intelligence? * How can artificial intelligence be utilized and created in an ethical way? === More Discussion questions and essay ideas=== * Compare prompt engineering to loop engineering to graph engineering. ** Compare and contrast these different modalities. ** When and how can each one be useful? * What are some different notable agentic harnesses and what the pros and cons of some of the major ones? ==Educational/research/learning prompt ideas== * Create pages for an educational wiki that demonstrate how agentic AI can be utilized to build out an educational wiki. * What goals should an educational wiki have. That is, for a wiki meant for learning, teaching, and research, how should such collaborative content actually benefit the readers and collaborators. * How could AI generated video be utilized for learning, teaching, and research? * What are economic aspects of universities that readers and creators on distributed educational wikis can possibly learn from or benefit from? * How those who utilize agentic AI benefit from agentic AI utilize decentralized wikis to have their agent collaborate on their behalf and on behalf of humanity? * How could Hermes agents collaborate to advance educational wikis - wikis that are meant for learning, teaching, and research? === Wikipedia === ==== Wikipedia Readings/Pages ==== {{colbegin|4}} : {{w|Explainable artificial intelligence}} : {{w|Artificial intelligence}} : {{w|Accelerating change}} : {{w|Quantum computing}} : {{w|Artificial brain}} : {{w|Artificial general intelligence}} : {{w|Mind uploading}} : {{w|Technological singularity}} : {{w|Intelligence explosion}} : {{w|Superintelligence}} : {{w|AI takeover}} : {{w|Existential risk from artificial general intelligence}} : {{w|Friendly artificial intelligence}} : {{w|Computronium}} : {{w|Distributed artificial intelligence}} : {{w|Machine learning}} : {{w|Philosophy of artificial intelligence}} : {{w|Hardware for artificial intelligence}} : {{w|Weak AI}} : {{w|Deep learning}} : {{w|History of artificial intelligence}} : {{w|Synthetic intelligence}} : {{w|Template:Natural language processing}} (template) : {{w|List of datasets for machine-learning research}} : {{w|Fine-tuning (machine learning)}} : {{w|Convolutional neural network}} : {{w|Natural language processing}} : {{w|Transformer (machine learning model)}} : {{w|Prompt engineering}} : {{w|PaLM}} : {{w|GPT-4}} : {{w|Auto-GPT}} : {{w|BLOOM (language model)}} : {{w|Hugging Face}} : {{w|Hallucination (artificial intelligence)}} : {{w|AI safety}} : {{w|AI alignment}} : {{w|Existential risk from artificial general intelligence}} : {{w|Template:Existential risk from artificial intelligence}} (template) : {{w|Ethics of artificial intelligence}} : {{w|Generative artificial intelligence}} : {{w|Artificial general intelligence}} : {{w|Artificial intelligence art}} : {{w|Music and artificial intelligence}} : {{w|Generative pre-trained transformer}} : {{w|Regulation of artificial intelligence}} : {{w|Wikipedia:Large language models}} (policy) : {{w|Language model}} : {{w|LLaMA}} {{colend}} ===== Wikipedia Categories ===== : {{wc|Artificial intelligence}} :: {{wc|Artificial neural networks}} ::: {{wc|Neural network architectures}} :: {{wc|Machine learning}} ::: {{wc|Datasets in machine learning|... datasets}} ::: {{wc|Deep learning}} :::: {{wc|Deep learning software|... software}} :: {{wc|Generative artificial intelligence|Generative AI}} :: {{wc|Applications of artificial intelligence}} ::: {{wc|Large language models}} ::: {{wc|Natural language processing}} ::: {{wc|Speech recognition}} :: {{wc|Existential risk from artificial general intelligence}} :: [[w:List of artificial intelligence projects|AI projects]] ===External === * [https://www.youtube.com/watch?v=Jy3zjXK4ao4 Google Just Put an A.I. Brain in a Robot (Research Breakthrough)] (video) ==See also== [[Image:Artificial Intelligence & AI & Machine Learning - 30212411048.jpg|thumb|300px|right]] {{colbegin|3}} * [[Robotics Projects]] * [[Introduction to Robotics]] * [[Arduino Robot]] * [[Automation]] * [[History of artificial intelligence]] * [[Anthropomorphic Robotics Project]] * [[Large language models]] * [[Robot car]] * [[Automatic Control and Robotics]] * [[Open-source domestic robots]] * [[Portal:Artificial intelligence]] * [[Center_for_Corporate_Auditing,_Responsibility_and_Management_policy_Authoring/Artificial_intelligence_eschewal_policy|Artificial intelligence eschewal policy]] * [[Wikiversity:Artificial_intelligence]] (Policy) {{colend}} ==External links== * [https://www.elementsofai.com/ Course on Elements of AI] * [https://www.reddit.com/r/artificial/ Discuss artificial intelligence on Reddit] * https://tldr.tech/api/latest/ai [[Category:Artificial intelligence| ]] 0ayunl1th277900o3yz0itnjcf1q591 0 109924 2835510 714045 2026-10-02T13:30:27Z Ega5l 3111996 Gea5l 2835510 wikitext text/x-wiki *'''Original language''': English *'''Original form and meaning''': ''gay'', adj./noun, 'homosexual' (Note: If the status is not specifically indicated then the word is stylistically neutral and generally used; if earlier meaning and status equals current use the former may be expressed by writing "dito". Cf. also the project [[ELiX Wiki:Projects/EuroLex|guidelines]].) <table border=1> <tr> <td width=10%>'''Language''' <td width=15%>'''Form''' <td width=10%>'''Date of Borrowing (and Obsolescence) ''' <td width=25%>'''Current Meaning and Status''' <td width=25%>'''Earlier Meanings and Statusses''' <td width=15%>'''Source''' <tr> <td>Catalan <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Croatian <td>''gay'' [=English pronunciation], noun <td>end20c <td>meaning: 'homosexual'; status: restricted use: modish <td>dito <td>DEA* <tr> <td>Czech <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Danish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Dutch <td>''gay'' [ge:j], adj. <td>1980s <td>meaning:'homosexual'; status: restricted to slang, modish <td>dito <td>DEA* <tr> <td>English <td>''gay'' <td>12c <td>1.) 'homosexual', 2.) 'carefree and merry', 3.) 'brightly coloured' <td>[http://www.etymonline.com/index.php?search=gay&searchmode=none] <td>Collins English Dictionary (4th edition) <tr> <td>Estonian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Finnish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>French <td>''gay'' [gε], adj. <td>1960s <td>meaning: 'homosexual'; status: restricted use: modish <td>dito <td>DEA* <tr> <td>Frisian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>German <td>''gay'' [gei, ge:], adj. <td>? <td>meaning: 'homosexual'; status: restricted use: modish, less frequent than ''schwul'' <td>dito <td>DEA* <tr> <td>Hungarian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Irish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Italian <td>''gay'' [gεi], noun <td>1970s <td>meaning: 'homosexual'; status: fully accepted but still marked as English, equally frequent as ''omosessuale'' <td>dito <td>DEA* <tr> <td>Latvian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Lithuanian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Maltese <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Norwegian <td>''gay'' [= Engl. pron] <td>1970s <td>meaning: 'homosexual'; status: restricted to colloquial and modish use, less frequent than ''homo, homofil'' <td>dito <td>DEA* <tr> <td>Polish <td>''gej/gay'', noun <td>end20c <td>meaning: 'homosexual'; status: restricted to colloquial use <td>dito <td>DEA* <tr> <td>Portuguese <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Rumantsch <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Slovak <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Slovenian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Spanish <td>''gay/gai'' [gei/gai], noun <td>1970s <td>meaning: 'homosexual'; status: fully accepted but still marked as English, less frequent than ''homosexual'' <td>dito <td>DEA* <tr> <td>Swedish <td>''...'' <td>... <td>'...' <td>'...' <td>... </table> *'''Annotations''': *DEA = ''Dictionary of European Anglicisms'' by Manfred Görlach (2001), Oxford: OUP. *'''Information on Other Languages''': '''Icelandic''': ''gay'' [= Engl. pron.], adj.; end20c; meaning: 'homosexual'; status: restricted to slang, less frequent than ''hýr''.''' Romanian''': ''gay/ghei'' [= Engl. pron.], noun; 1990s; meaning: 'homosexual'; status: rare, less frequent than ''homosexual''. '''Russian''': ''geĭ'', noun; end20c; meaning: 'homosexual'; status: restricted to colloquial use. '''Bulgarian''': ''geĭ'',noun; 1990s; meaning: 'homosexual'; status: restricted *ega5 * * * * * * * *d to journalese, modish, less frequent than ''khomoseksualist, pederast, pedi''. '''Greek''': ''gei/gay'', adj.; meaning: 'homosexual'; status: restricted to [[Category:EuroLex]] col [[Category: Anglicism]] loquial and modish use, journalese. , iyhxvua8c8vd9d2lswj89by9q2w0yl2 2835511 2835510 2026-10-02T13:31:40Z Ega5l 3111996 /* */ 2835511 wikitext text/x-wiki *'''Original language''': English *'''Original form and meaning''': ''gay'', adj./noun, 'homosexual' [[Ahmeddye|Ahmed]](Note: If the status is not specifically indicated then the word is stylistically neutral and generally used; if earlier meaning and status equals current use the former may be expressed by writing "dito". Cf. also the project [[ELiX Wiki:Projects/EuroLex|guidelines]].) <table border=1> <tr> <td width=10%>'''Language''' <td width=15%>'''Form''' <td width=10%>'''Date of Borrowing (and Obsolescence) ''' <td width=25%>'''Current Meaning and Status''' <td width=25%>'''Earlier Meanings and Statusses''' <td width=15%>'''Source''' <tr> <td>Catalan <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Croatian <td>''gay'' [=English pronunciation], noun <td>end20c <td>meaning: 'homosexual'; status: restricted use: modish <td>dito <td>DEA* <tr> <td>Czech <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Danish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Dutch <td>''gay'' [ge:j], adj. <td>1980s <td>meaning:'homosexual'; status: restricted to slang, modish <td>dito <td>DEA* <tr> <td>English <td>''gay'' <td>12c <td>1.) 'homosexual', 2.) 'carefree and merry', 3.) 'brightly coloured' <td>[http://www.etymonline.com/index.php?search=gay&searchmode=none] <td>Collins English Dictionary (4th edition) <tr> <td>Estonian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Finnish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>French <td>''gay'' [gε], adj. <td>1960s <td>meaning: 'homosexual'; status: restricted use: modish <td>dito <td>DEA* <tr> <td>Frisian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>German <td>''gay'' [gei, ge:], adj. <td>? <td>meaning: 'homosexual'; status: restricted use: modish, less frequent than ''schwul'' <td>dito <td>DEA* <tr> <td>Hungarian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Irish <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Italian <td>''gay'' [gεi], noun <td>1970s <td>meaning: 'homosexual'; status: fully accepted but still marked as English, equally frequent as ''omosessuale'' <td>dito <td>DEA* <tr> <td>Latvian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Lithuanian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Maltese <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Norwegian <td>''gay'' [= Engl. pron] <td>1970s <td>meaning: 'homosexual'; status: restricted to colloquial and modish use, less frequent than ''homo, homofil'' <td>dito <td>DEA* <tr> <td>Polish <td>''gej/gay'', noun <td>end20c <td>meaning: 'homosexual'; status: restricted to colloquial use <td>dito <td>DEA* <tr> <td>Portuguese <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Rumantsch <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Slovak <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Slovenian <td>''...'' <td>... <td>'...' <td>'...' <td>... <tr> <td>Spanish <td>''gay/gai'' [gei/gai], noun <td>1970s <td>meaning: 'homosexual'; status: fully accepted but still marked as English, less frequent than ''homosexual'' <td>dito <td>DEA* <tr> <td>Swedish <td>''...'' <td>... <td>'...' <td>'...' <td>... </table> *'''Annotations''': *DEA = ''Dictionary of European Anglicisms'' by Manfred Görlach (2001), Oxford: OUP. *'''Information on Other Languages''': '''Icelandic''': ''gay'' [= Engl. pron.], adj.; end20c; meaning: 'homosexual'; status: restricted to slang, less frequent than ''hýr''.''' Romanian''': ''gay/ghei'' [= Engl. pron.], noun; 1990s; meaning: 'homosexual'; status: rare, less frequent than ''homosexual''. '''Russian''': ''geĭ'', noun; end20c; meaning: 'homosexual'; status: restricted to colloquial use. '''Bulgarian''': ''geĭ'',noun; 1990s; meaning: 'homosexual'; status: restricted *ega5 * * * * * * * *d to journalese, modish, less frequent than ''khomoseksualist, pederast, pedi''. '''Greek''': ''gei/gay'', adj.; meaning: 'homosexual'; status: restricted to [[Category:EuroLex]] col [[Category: Anglicism]] loquial and modish use, journalese. , swpz3vsjg37ipva8km63yimpcpaipwf 0 139384 2835611 2835326 2026-10-03T04:51:59Z Young1lim 21186 2835611 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.20261002.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261002.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]] 44v8sk73vurvbpih9196i56yte2zpwh 2835614 2835611 2026-10-03T04:53:05Z Young1lim 21186 /* Adder */ 2835614 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.20261003.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261002.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]] pe549z3ccgesecic4fk4mbqofnk875i 2835617 2835614 2026-10-03T04:55:02Z Young1lim 21186 /* Adder */ 2835617 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.20261003.pdf|B]], [[Media:VLSI.Arith.2C.CLA.20261003.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]] 15uttxecbotdy06c1yj3e4r0jfxu0t5 2 152101 2835555 1436498 2026-10-02T18:56:06Z TommyG 332141 ([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Cat 5 cable.jpg]] → [[File:Cat3 Ethernet TP cable.jpg]] [[c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name) · not cat 5 2835555 wikitext text/x-wiki =Finish Description= =Conceive= 1. '''Market/Customer Needs''' To help someone open their lock, if they forget their combination. 2. '''Initial target goals''' To make an automatic combo lock pick. To have the hinge fold the lock pick to a resting position so that a new lock can be put in. 3. '''Project cost and schedule''' We all ready have all of the parts needed. 4. '''Be able to open a Masterlock without knowing the combination''' 5. '''Ability to easily remove abandoned locks from public lockers''' *Replace bolt cutters *Locks still useable after removal *Retrieve forgotten combination Original project by students at Olin College: http://boingboing.net/2011/03/10/combination-lock-cra.html Step by Step guide to how a masterlock can be opened: http://www.markedwardcampos.com/files/gimgs/13_mcamposfinal.png =Design= *Tutorial of Combo Lock Pick Method Found **[http://www.wikihow.com/Crack-a-%22Master-Lock%22-Combination-Lock Wikihow] steps for cracking a Masterlock Combo Lock. **[http://hackaday.com/2009/10/06/how-to-crack-a-master-lock/ Hackaday] step by step guide for Masterlock. === Oldest Body Design === <Gallery> File:MechanicalComboLockPick.jpg|The system </Gallery> ==== Material List ==== -1 Stepper Motor -2 Arduinos, stacked -1 Masterlock brand combination lock -1 small piece of wood (to serve as a base) -1 pull tie (to fasten the motor to the wood) -1 plastic connector piece (unknown origin, part of a pile of random plastic scrap) === Oldest Wiring Design === <Gallery> File:Lockpick poster.jpg|Everything connected and working. </Gallery> ==== Material List ==== 1. A [http://www.digikey.com/product-detail/en/G5G-1A-DC24/Z3105-ND/1815577?WT.mc_id=PLA_1815577&gclid=CJ7K29an9LICFROe4AodGxcAwg relay] 2. A [http://www.alldatasheet.com/datasheet-pdf/pdf/108285/ROHM/2SD1765.html 2SD1765] transistor 3. A 22K Ohm resistor === Original Body Design === The first thing that we did was screwed two blocks off wood to the door hinge. However this became problematic because the hinge did not extend to a full 90 degrees angle. To solve this we cut the wood at a angle to get the hinge to extend to 90 degrees. The next problem was extending the wood back because the platform of the hinge was getting in the way. To fix this we cut out a block off wood where this hinge was hitting. After doing this the hinge could extend in and out with out a problem. <Gallery> File:Front view of combo lock pick.png|The lock and puller have been mounted on the back piece of wood. File:Side of project.png|The lock and puller have been mounted on the back piece of wood. File:Extended view.png|The hinge allows the lock pick to be folded into a working and non-working position. </Gallery> ==== Original Gear Structure ==== The next thing we did was make gear structure that will link up the the gear on the motor, and be attached to the rubber stopper. All of these pieces must spin together to work. So they all have to connected to each other. *The middle gear is used to link to the gear on the motor. *The washer is what will be put into the plastic holder. * The plastic holder will hold up the gears. <Gallery> File:Gears close up.png|Gears made to link with the one attached to the motor. File:Back of gears.png|A bolt runs through the gears that is attached to the rubber holder. File:Gears finished.png|The gears after the washer is put in. </Gallery> The project needed to be raised up to prevent the hinge from making it off balance. Next we drilled a hole into the bottom board and glued a plasic holder in. We are using this to try and hold up the gear structure. <Gallery> File:Hole drilled into bottom block of wood.png|This hole is drilled to put the plastic in that will be holding up the gears. File:Plastic is glued into place.png|The holder is glued into the hole. File:Height added to the lock pick.png|Blocks of wood are screwed into the bottom top add stability to the project. Also the hinge can now be put back without moving the whole project. </Gallery> === Body Re-Designed === <Gallery> File:Side view of wood under the hinge.JPG|The wood pushes the hinge up to extend the hinge further to try and get it to a 90 degree angle . File:Bottom view of wood under the hinge.JPG|The wood pushes the hinge up to extend the hinge further to try and get it to a 90 degree angle. File:Both boards attached to hinge.JPG|Both boards attached to hinge. File:Board with lock.JPG|The board with the lock is attached to the upright board. The upright board isn't all the way to 90 degrees but the nuts on the back of the lock board push it out to a 90 degree angle. </Gallery> <Gallery> File:Wood boards are added underneath.JPG|Wood boards are added underneath to hold up the project, and to keep it balanced. File:Underneath View.JPG|Underneath view of the project with new boards added to stabilize. File:Turn-able Wood Blocks.JPG|Turn-able blocks are added to prevent the project from falling back when the lock is being pulled up. File:Blocks turned to the side.JPG|Blocks turned to the side to allow the lock board to but pulled back, but the blocks are too big so the board can not be reclined. File:Side view of the turn-able blocks.JPG|Side view of the turn-able blocks. File:Upright cut blocks.JPG|The turn able blocks are cut to allow the lock board to recline. Right now they are upright to hold the board up. File:Sideways cut blocks.JPG|The turn able blocks are cut to allow the lock board to recline. File:Lock board reclined.JPG|Lock board reclined. </Gallery> ==== Gear Structured Re-Designed ==== *The back gear connects the the gear on the motor. *The washer is used so that the gears connect to it, so the inside of the bearing moves while the outside is glued in place. *The middle gear is used to connect to the bearing. *A bolt is running through the middle to hold everything together. *A metal hook is bent to hold the bearing in it. <Gallery> File:Bearing to hold up gears.png|Bearing that will be used to hold up with the metal stand, while the middle can still spin File:Gears interlocked.png|Gears interlocked File:New Gear structure.png|Gears all put together File:Gear that will hook up to Motor gear.png|Gear that will hook up to Motor gear File:Bearing on Gear.png|Bearing on Gear File:Screw Through Gear.png|Screw Through Gear </Gallery> <Gallery> File:Metal Holder.png|Metal holder that will hold up gears File:Metal Stand.png|Metal that will hold up gears File:Gears Held Up.png|Gears held up against lock </Gallery> *We decided to use a board with a bearing in it because this is more stable then gluing a bearing onto a metal hook. *The board was cut and screwed into place. *A gear that is the same size as the hole in the bearing is put in it. The other gears fits around the fist gear and a bolt runs through the rubber stopper and all of the gears. *Washers are placed on both sides of the bearing, and on both ends off the gears,where the nuts are. A problem with this design is that when the gears are turned the nuts loosen and tighten. This is why the washers are added. They are supposed to prevent this. Another problem is that the screw head inside the rubber is too big. This is a problem because the rubber can not grip the lock anymore. To solve this the bolt head is filed down. <Gallery> File:Bearing inside board.JPG|Bearing inside board. This can be used to hold up the gears, but keep them turning. File:Side view of bearing in a board.JPG|Side view of bearing in a board. File:Gear that fits in bearing.JPG|Gear that fits in bearing. File:Side view of gear in bearing.JPG|Gear in bearing. File:Back of gear in bearing.JPG|Gear in bearing. File:Bearing board screwed in.JPG|Bearing board screwed in. File:Side view of bearing board screwed in.JPG|Side view of bearing board screwed in File:Bolt that is runs through all of the gears.JPG|Bolt that is runs through all of the gears. File:Screw head is filed down.JPG|The head of the screw is filed down so that the rubber piece can properly hold the lock. The head of the screw was getting inn the way. File:Gears all put together.JPG|Gears are all put together but when they are tuned the bolts loosen up. File:Gears all put together.JPG|I might add washer to try and prevent the nuts from loosening. File:Washers on gears.JPG|Washers are added to try and keep the nuts from spinning and therefor loosening or tightening the gears. File:Gears with washers.JPG|Gears assembled and linked to lock. </Gallery> ==== Final Working Gear Design ==== The final design for the project consists of a large, stable base section, raised off the ground to allow the mounting of a two-position hinge. This hinge is used to direct and control movement of an upright section, upon which are mounted the solenoid, lock, and a PVC that holds the lock in place. The motor and shaft assembly are mounted in a fixed location in front of where the lock will be in an upright position, allowing the lock and rubber stopper to engage via pressure. Behind the upright section, two small wooden blocks on pivots can be swung upwards to hold the assembly in place or downwards to allow the assembly to recline. Reclining the assembly removes the lock from in front of the motor assembly and into a position where it can be removed and replaced. The Arduino and wiring for the motor and solenoid will be located off to the side, at this point unmounted so as not to interfere with the operation or or further alteration of any other parts. <gallery> File:New Gear Mount Top View.jpg|top view: A flat head screw was used to stop interference with the lock when trying to mount on the lock. File:New Gear Mount.jpg|Better view of the assembly, shows how the screw is threaded through the rubber and gear. File:New Mount.jpg|the screw is threaded all the way through the rubber and gear for a secure connecting. File:Better view of the gear mount.jpg|this is the final mounting process, everything moves freely with no interference from other parts. </gallery> ==== Videos of Gears Spinning ==== Here is a video to show the improvement we've made after brainstorming for days: *Before Video: http://www.youtube.com/watch?v=TGGgvnxqCQo&feature=youtu.be<gallery> File:Wooding mount.jpg </gallery> *After Video: http://www.youtube.com/watch?v=QHucV7Zd2ks&feature=youtu.be<gallery> File:New Mount.jpg </gallery> ===Arduino Code and Wiring=== Reading code written by last years' teams provided a starting point for writing our own code for using the Arduino and motor shield with the provided stepper motor. The first attempt at writing code for the project built off of the old teams' work, altering the step size and timing to match with the stepper motor provided. We attempted a [https://en.wikiversity.org/wiki/User:Jdittman8063/ENES-100/project_1/Quick_Test Quick Test] of the code by attempting to run a shortened version of it. By attempting to loop through 16 possible combinations, it was found that the code and motor shield wiring based on previous teams' efforts had multiple errors. The [http://arduino.cc/en/uploads/Main/arduino_MotorShield_Rev3-schematic.pdf Wiring Diagram] for the motor shield was also helpful in determining how to properly wire the motor to the motor shield. At this time we located a [http://www.instructables.com/id/Arduino-Motor-Shield-Tutorial/ Tutorial] on how to use the motor shield properly. The wiring for the motor was also called into question and a simple test showed that it was indeed wired improperly. This led to stepping back to start with the code and wiring of the motor, Arduino, and motor shield. At this time we are working on writing new code to operate the stepper motor to follow the implied algorithm for cracking the lock. [https://commons.wikimedia.org/wiki/File:Matching_Motor_Wires.ogv Video] of the LED turning on to show a matched pair of wires from the motor. *Tutorial of Combo Lock Pick Method Found **[http://www.wikihow.com/Crack-a-%22Master-Lock%22-Combination-Lock Wikihow] steps for cracking a Masterlock Combo Lock. **[http://hackaday.com/2009/10/06/how-to-crack-a-master-lock/ Hackaday] step by step guide for Masterlock. === Blink Program === Material: *Arduino *Power Switch Tail *Sensor *Solenoid How to get to the blink program: *Engshortcut *Arduino *Ports *USB serial port (Com 14) *Blink Program <Gallery> File:Arduino Blink File.JPG|This is a file that is already made on the Arduino program on the school computer. It is the blink program. File:Blink File.JPG|I changed the blink file to activate on the arduino when the orange wire is in port 5, not 13. Because port 13 is needed for the motor program. File:Arduino Connected to computer.JPG|Arduino linked to computer to send it the program to run the solenoid. File:Wires Coming off of the power cable.JPG|The power cable connects to the solenoid and to the arduino. The red wire is the ground. The orange wire is what runs the program. File:Wires in Arduino.PNG|Red goes to the ground, and when the orange goes into the correct port it shoots up the solenoid. File:Solenoid Pulled Up.PNG|The solenoid is told by the arduino to contract up. File:Solenoid Switch.JPG|This switch could be used to mount on top of the solenoid to tell it when it hits it to stop the program because the combination has been solved. </Gallery> Solenoid activated using Port 13 on the Blink program: http://www.youtube.com/watch?v=3N0Egcpq_8c&feature=youtu.be We moved the Solenoid activation to port 5 to see what happens: http://www.youtube.com/watch?v=w8uWqiPF8JI&feature=youtu.be Arduino connecteed to the Solenoid: http://www.youtube.com/watch?v=qMwK1ov807k&feature=youtu.be <Gallery> File:Power Source Plugged Into Solenoid.JPG|Power source plugged into solenoid File:Power Source Back.JPG|Power Source screwed in to the upright board. </Gallery> Video of the Combo lock relining back to change out a lock, then coming back into position: http://www.youtube.com/watch?v=cFAW6Mqz9Vo&feature=youtu.be After getting the Blink program, we connected the arduino to the computer and uploaded the program. if done properly the led on the board should be flashing on/off every second. next, we Plugged in the Power Switch Tail and took hold of the two wire. the red wire will be plugged in to the port labeled ground and orange wire will go into port 13 which will active the solenoid. <gallery> File:Power Switch Tail.jpg|The Power switch tail was move to the back of the board to leave space of the arduino and motor shield in the front. File:Powering the Solenoid.jpg|power is delivered to the solenoid through the Power Switch Code, this is one of the reason why the switch tail is mounted to the back of the board. File:Figuring out Mounting Position..jpg|There was enough space for the arduino and shield after moving the tail switch to the back of the project board. </gallery> After playing around with the arduino program, we was able to modify the current blink program to help the solenoid snap a lot better by decreasing the High output to 50ms and increasing the Low output to 2000ms. this gave the solenoid a faster snap then a longer release when the solenoid is active. The modify Blink code that gave the solenoid a more Natural snap: http://www.youtube.com/watch?v=JkO884DU_N4&feature=youtu.be === Mounting the Arduino === This is a broken arduino that we mounted before the one that will work with our project. We first mounted this one to ensure that we had a way to put it on the base first without attemping to put on the working arduino. <Gallery> File:Rubber pieces that will be under the arduino.JPG|These rubber pieces go under the arduino, and around the screws. File:Rubber pieces on the board.JPG|This is showing where the rubber pieces will go on the wood. File:Top view of arduino screwed into the board.JPG|Top view of arduino screwed into the board. File:Rubber under the arduino.JPG|Rubber under the arduino. File:Side view of rubber under the arduino.JPG|Side view of rubber under the arduino File:Rubber under the top two screws.JPG|Rubber under the top two screws. File:Screws with rubber through the arduino.JPG|The screws run through the arduino with the rubber on the bottom to keep the arduino from coming into contact with the wood. </Gallery> === Extending Wires === <Gallery> File:Ground wire does not reach the arduino.JPG|Ground wire does not reach the arduino. File:Wires connected before they are extended.JPG|The wires need to be extended because when the solenoid is pulled back the wires are too short to stay connected to the arduino. File:Power wire is extended.JPG|Power wire is extended by soldering an extra wire to it. File:Heat wrap on power wire.JPG|After being soldered the power wire is wrapped in a heat shrink wrap. File:Ground wire is extended.JPG|Ground wire is extended by soldering. File:Ground wire is extended again.JPG|Ground wire is extended again by soldering. </Gallery> Video of the wires being to short to stay connected while the solenoid and lock are pulled back: http://www.youtube.com/watch?v=RnwxFWSSMJI&feature=youtu.be Video of the solenoid and lock being pulled back, the wires stay in place: http://www.youtube.com/watch?v=1t0uR5_pfHs&feature=youtu.be <Gallery> File:Hole drilled for wires.JPG|a hole is drilled to channel the wires through. File:Wires running along bottom.JPG|Wires running along bottom. This makes them hidden from sight. File:Wires coming through the hole.JPG|Wires coming through the hole, but they aren't long enough. File:Wires go underneath the wood.JPG|Wires go underneath the wood. File:Wires soldered.JPG|Wires extended to reach the arduino. File:Wires coming through the bigger hole.JPG|wires are now long enough. File:Wires connected to the arduino.JPG|Wires coming through the hole and connecting to the arduino. File:Wires running under motor.JPG|Wires running under motor. File:Hook 1 holding wires in place.JPG|Hook by the hinge is holding the wires in place as they come in from the power block. File:Hook 2 holding wires in place.JPG|Second hook is placed by the the hole under the arduino. </Gallery> the regular wire on the Optical switch was very weak. We soldered new wire on to exciting wire to increase it strength and durability. this will allow for a better connection point for the the arduino. after these connection as been made, this should give better contact points on the arduino. <gallery> File:Switch3.jpg|The clips are used to hold up your wires when your soldering so you won't burn yourself. File:Switch5.jpg|after you have soldered, you must us heat shrink tubing to secure your solder. File:Switch7.jpg|Wire for the switch needs to be solder to work with the arduino. thicker wires are good for this, for they can hold there shape for a more easy connection. </gallery> We were giving a mechanical switch work on. We were giving the task to create an arduino code that will work with the switch. but, before we can do that we must first solder a resister on the wire to make sure it not taking in to much power, and a wire that's more compatible with the arduino. <gallery> File:Heat shrink tubing.jpg| Heat shrink Tubing is used to secure wires that you striped and soldered together giving it a strong insulator. File:Mechanical Switch.jpg|soldering longer wire to be able to work with an arduino. File:Mechanical Switch-0.jpg| Heat shrink Tube was used to protect the the resister and wire soldering points when they have cooled down. File:Mechanical Switch-1.jpg|Wire used to receive power must be solder carefully to make sure that there will be no shorting of the wire during operation. </gallery> === Solenoid Hook === <Gallery> File:Thicker hook.JPG|A thicker hook is needed connect to the lock because the solenoid would bent the thinner one out of shape. File:New thicker hook.JPG|thicker hook connected to lock. </Gallery> Video of the thicker hook connected to the solenoid: http://www.youtube.com/watch?v=ZnLiqRuQBjM&feature=youtu.be === Re-designing Gear Structure === <Gallery> File:Motor Connected to gear.JPG|the motor needs to connect to the gears by moving forward because there is a line hat runs through the middle of the gear. File:Gears moved forward.JPG|The bearing board is moved forward so that the gears can be pushed back so the motors gear can hook up correctly. File:New screw placed through the gear..JPG|A thinker screw is placed through the gears and rubber piece because the old screw started to slip. File:Back view of new screw.JPG|The new screw is shorter so it doesn't poke out the back. </Gallery> === Mounting Stepper Motor and Arduino === Video of gears being spun by hand: http://www.youtube.com/watch?v=ETb02RsUdVU&feature=youtu.be <Gallery> File:Arduino mounted.JPG|The arduino with the programs is mounted to the project. File:Side view of arduino.JPG|Side view of arduino mounted. File:Wooden block that will hold up motor.JPG|Wooden block that will hold up motor File:Wooden Block screwed in.JPG|The wooden block that will hold up the motor is screwed in. File:Motor on top the block.JPG|The wood block will hold up the motor so the gear doesn't scrap the wood. File:Metal clamp.JPG|A metal clamp is used to tighten up motor. File:Motor held in place.JPG|The motor is held in place by a metal clamp that is tightened. </Gallery> Making the final mount for the motor, the 2in x 2in block is where the motor will sit so it will be next to secondary gear for the rubber mount. <gallery> File:Final mounting of motor 12.jpg|scrap wood was used for the gap in the secondary board in case of flex. File:Final mounting of motor.jpg|i was matching the tooth of each gears together before paining where to mount the motor on the board. File:Final mounting of motor 2.jpg|getting a rough of where the motor will be sitting on the project board. the motor must be close enough to make connect with the second gear but not obstruct any wire around it. File:Final mounting of motor 13.jpg|a 2in x 8in board was found in the ENES workshop, this will be cut into a block for witch the motor will sit on. File:Final mounting of motor 3.jpg|found a 2in x 8in board in the ENES workshop that can be cut to 2in x 2in block as a base in witch the motor will sit on during operation. make sure the board is secure in the clamp. File:Final mounting of motor 4.jpg|Make your measurement to 2in x 2in and then proceed to cutting the board. make sure the board is secure in the clamp. File:Final mounting of motor 6.jpg|the 2in x 2in block is hold in place by two two inches wood screw with a flat head. make sure to leave space for further modification. File:Final mounting of motor 7.jpg|a hinge that we found laying around in the ENES workshop was going to be used as a mounting bracket for the motor. File:Final mounting of motor 5.jpg|the hinge was bend to fit the shape of the motor so that i can secure it to the project board with screws. the problem is we had a hard time bending the hinge to shape. File:Final mounting of motor 9.jpg|a metal screw bracket was found in a bucket in the ENES workshop, so we made to small holes in the center of the bracket that will allow use to secure the bracket to the frame. File:Final mounting of motor 8.jpg|After making the pilot hole for the screw, secure the whole thing down with wood screws, this may be hard because the wood screw is made out of a softer compound than the aluminum bracket so it might strip screw. be careful. File:Final mounting of motor 10.jpg|before tightening down the motor, make final adjustment for a more better connection. File:Final mounting of motor 11.jpg|secure everything down and you are finish. </gallery> === Holding the Project together === <Gallery> File:Hooks on the main board.JPG|Hooks will be used to hold a string to keep the project from moving so much. File:Hooks on top and bottom.JPG|Hooks will hold the project together by the tension of a string. File:Fishing line used.JPG|Fishing line is tied and used to hold the project together. File:Steel cable connected to hook.JPG|Steel cable connected to hook. File:Cable_hooks.jpg|Both ends of the steel cable with hooks attached. File:Steal cable wrapped around hooks.JPG|The wire is wrapped around the hooks to hold the project together. </Gallery> strengthening the Structural integrity of the project board by replacing the fishing line with steel cables. <gallery> File:Increasing structural integrity.jpg|fishing line was first used to see if we can increase the structural strength of the frame. File:Increasing structural integrity 2.jpg|the fishing line was strung from the top hooks to the base hook for more added strength, the problem is the fishing line lacks strength to keep the board from collapsing. File:Increasing structural integrity 3.jpg|the fishing line was scraped because it lack strength, and was replaced with a steel cable that was found laying around in the ENES workshop. File:Increasing structural integrity 6.jpg|you'll need this crimping tool in order to crimp together the hook and steel cable. File:Increasing structural integrity 4.jpg|yellow large crimps where used to hold the steel cable on either end, hooks where also crimped on the end of each steel cable. this will be used for the larger hook on the project frame. File:Increasing structural integrity 5.jpg|the steel cable was wrapped around the bottom hooks and stretched to the top hooks to increase the strength of the project board structure. </gallery> === Coding for The Switch=== ======Code One====== When using this code to operate the machnical switch, we were not getting any response from the led after the switch was activated. this program was suppose to label pin 13 as led, and label pin 8 as switch pin. when the switch is activated the arduino reads the signal as high and turns the led on, else the led should be low. <gallery> File:This is the set up for both program one and two.jpg|This is the set up used for both program one, two, three. </gallery> <source lang="cpp">> int switchPin = 8; int ledPin = 13; void setup() { pinMode(switchPin, INPUT); pinMode(ledPin, OUTPUT); } void loop() { if (digitalRead(switchPin) == HIGH) { digitalWrite(ledPin, HIGH); } else { digitalWrite(ledPin, LOW); } } </source> ======Code Two====== For part two, the boolean was added to check the current state of the led. (boolean lastButton = LOW;) keeps track of watch the button did in the previous loop, and (boolean ledOn = false;) is used to keep track of the current state of the led. and they will help check things in the loop to help set thing to where there suppose to be. <gallery> File:This is the set up for both program one and two.jpg|This is the set up used for both program one, two, and three. </gallery> this is the youtube video showing the problem: http://www.youtube.com/watch?v=t-3kIytQPeQ&feature=youtu.be <source lang="cpp">> int switchPin = 8; int ledPin = 13; boolean lastButton = LOW; boolean ledOn = false; void setup() { pinMode(switchPin, INPUT); pinMode(ledPin, OUTPUT); } void loop() { if (digitalRead(switchPin) == HIGH && lastButton == LOW) { ledOn = !ledOn; lastButton = HIGH; } else { lastButton = digitalRead(switchPin); } digitalWrite(ledPin, ledOn); } </source> ======Code Three====== Code to works, but has a bit of a problem. the led turns of and on but you can still see a dim light coming from the led. In order to solve this problem we made our own function. the function is suppose to return a boolean. (boolean debounce(boolean last)) debounce is now a function with input last, this is now define as a that determine the current value with the remaining of the program. <gallery> File:This is the set up for both program one and two.jpg|This is the set up used for both program one,two, and three. </gallery> this is the youtube video showing the problem: http://www.youtube.com/watch?v=ib1kVmWDMzU&feature=youtu.be <source lang="cpp">> int switchPin = 8; int ledPin = 13; boolean lastButton = LOW; boolean currentButton = LOW; boolean ledOn = false; void setup() { pinMode(switchPin, INPUT); pinMode(ledPin, OUTPUT); } boolean debounce(boolean last) { boolean current = digitalRead(switchPin); if (last != current) { delay(5); current = digitalRead(switchPin); } return current; } void loop() { currentButton = debounce(lastButton); if (lastButton == HIGH && currentButton == HIGH) { ledOn = !ledOn; } lastButton = currentButton; digitalWrite(ledPin, ledOn); } </source> =====Hooking up a Switch to Arduino===== if your not confortable with writing you own code, the Analog in out Serial program can be found in the arduino software under (File, Example, Analog, and number three AnaloginoutSerial). this program works perfectly with the switch with now problem what so ever. <gallery> File:Diagram that must be followed.jpg|this the diagram that must be followed if you want to use the final code File:Following Wiring diagram.jpg|After following the wiring diagram, you should and up with this final wires in the right pins. </gallery> <source lang="cpp">> /* Analog input, analog output, serial output Reads an analog input pin, maps the result to a range from 0 to 255 and uses the result to set the pulsewidth modulation (PWM) of an output pin. Also prints the results to the serial monitor. The circuit: * potentiometer connected to analog pin 0. Center pin of the potentiometer goes to the analog pin. side pins of the potentiometer go to +5V and ground * LED connected from digital pin 9 to ground ======Code Final====== created 29 Dec. 2008 modified 9 Apr 2012 by Tom Igoe This example code is in the public domain. */ // These constants won't change. They're used to give names // to the pins used: const int analogInPin = A0; // Analog input pin that the potentiometer is attached to const int analogOutPin = 9; // Analog output pin that the LED is attached to int sensorValue = 0; // value read from the pot int outputValue = 0; // value output to the PWM (analog out) void setup() { // initialize serial communications at 9600 bps: Serial.begin(9600); } void loop() { // read the analog in value: sensorValue = analogRead(analogInPin); // map it to the range of the analog out: outputValue = map(sensorValue, 0, 1023, 0, 255); // change the analog out value: analogWrite(analogOutPin, outputValue); // print the results to the serial monitor: Serial.print("sensor = " ); Serial.print(sensorValue); Serial.print("\t output = "); Serial.println(outputValue); // wait 2 milliseconds before the next loop // for the analog-to-digital converter to settle // after the last reading: delay(2); } </source> After finishing the the coding for the switch, i help out with construction on the main board that will house all the component. hooks where added along with fishing wire to give the frame of the board more straight. <gallery> File:Screw grinding.jpg|the screw was grind down because it was in the way of the lock when we where trying to mount the rubber on the lock. File:Two hooks - Bottom.jpg|Two hooks are added to the bottom of the board and have a fishing line connecting both of them together. File:Two hooks - top.jpg|Two hooks are added to the top of the board and have a fishing line connecting both of them. File:Final product.jpg|the final product has four hooks that is connected with a fishing line to help with the flex of the board when the solinod is activated. </gallery> =New Switch= A new switch was acquired since the old switch did not work the way it was suppose to. the new switch is called the Vernier Photogate, it suppose to be more reliable then the switch that was salvaged from the an old hp printer. Materials Used For Project 3: *Cat 5 cable *Vernier Photogate *Lab Quest Tablet <gallery> File:Vernier Photogate.jpg|Vernier Photogate File:Lab Quest Tablet.jpg|Lab Quest Tablet File:Cat3 Ethernet TP cable.jpg|Cat 5 cable </gallery> <gallery> File:Cut Cat 5 Cable.jpg|you need to cut off one side of the cat 5 cable and leave the other side to be plug into the Vernier Photogate. File:Cut Cat 5 Cable 2.jpg|after cutting the cat 5 cable, you'll need to strip off each plastic coating to expose the copper wire. File:Cut Cat 5 Cable plug into Vernier Photogate..jpg|after making the cut, i pluged in the other end into the Vernier Photogate to get a visual idea of how it works. </gallery> *YouTube Video showing off the Verier Photogate: http://www.youtube.com/watch?v=v8rgyOOBPPU&feature=youtu.be this is a video showing how Verier Photogate works with the Lab Quest tablet combine. the lab Quest takes reading in a form of a graph. when you bridge the gap between the sensor it plots points on lab quest that can easy be downloadable and analysis. *Down below you will find links to webpages giving you more information between cat5e cable and data cable. i also found a wiring diagram explain what each wire does, all you'l need to do is match them us with each other and feed them to the arduino. the wiring diagram explains both the cat5 cable and the prog cable that comes with the photogate. http://www.vernier.com/support/sensor-pinouts/ https://en.wikipedia.org/wiki/Category_5_cable http://www.vernier.com/engineering/arduino/digital-sensors/photogates/ =Implement= The code currently being used to test the function of the project is shown below. It is able to properly operate the solenoid at the end of each combination attempt and will output the combination attempted at the end of each iteration. The next step for this is to devise an algorithm to more efficiently test combinations for the lock. <source lang="cpp">> //uses Adafruit Motor shield library #include <AFMotor.h> AF_Stepper motor(200, 2); int solenoid = 6; //set solenoid to digital pin 6 int first; int second; int third; void setup() { Serial.begin(9600); // set up Serial library at 9600 bps Serial.println("Complete test of stepper motor."); motor.setSpeed(120); // 120 rpm // initialize the digital pin as an output pinMode(solenoid, OUTPUT); digitalWrite(solenoid, LOW); } void loop() { //continue interations until lock is opened combinations(); } void combinations() { for (first = 1; first < 40; first++){ for (second = 1; second < 40; second++){ for (third = 1; third < 40; third++){ motor.step(first*5, FORWARD, DOUBLE); delay(250); motor.step(200, BACKWARD, DOUBLE); motor.step(second*5, BACKWARD, DOUBLE); delay(250); motor.step(third*5, FORWARD, DOUBLE); delay(250); digitalWrite(solenoid, HIGH); // turn the solenoid on (HIGH is the voltage level) delay(20); // wait for a 0.02 seconds digitalWrite(solenoid, LOW); // turn the solenoid off by making the voltage LOW delay(100); // wait for a tenth of a second Serial.print("Combination attempted is "); Serial.print(first); Serial.print(", "); Serial.print(second); Serial.print(", "); Serial.print(third); Serial.print("."); //finish with three full forward rotations motor.step(600,FORWARD, DOUBLE); delay(1000); } } } } </source> =Operate= The project as it is currently assembled is shown below from both a top and side view. When testing the entire project together, it was found that activating the solenoid for too long of a period at once was causing the PVC piece to come unglued from the wooden structure. <Gallery> File:Top_view_of_lock_picker.jpg File:Side_view_of_lock_picker.jpg File:Lock_holder_unglued.jpg </Gallery> As the two videos below show, each part of the project functions individually, but the motor's primary gear does not mesh with the second gear properly. This results from a difference in tooth shape and pitch between the two gears. This will be addressed in the future. <Gallery> File:Project_test1.ogv File:Motor_test2.ogv </Gallery> =Next Steps= To push the project forward: ===Arduino=== *Build off of the code provided by the motor shield tutorial found here. http://www.instructables.com/id/Arduino-Motor-Shield-Tutorial/ *Build a switch that will work with both the Arduino and motor shield that will stop and start the program. *Verify proper power source for the motor and solenoid, to eliminate need to draw power from a computer. *Explore options for mounting and housing the Arduino and motor shield in a way which protects them from drops, dings, etc. ===Body=== *Make a smaller, sturdier, compact body so that the lock pick can be moved around easier 0vdabr6ksnn75lgpwvax3lzh68qbwwv 2 152357 2835556 1107694 2026-10-02T18:56:09Z TommyG 332141 ([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Cat 5 cable.jpg]] → [[File:Cat3 Ethernet TP cable.jpg]] [[c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name) · not cat 5 2835556 wikitext text/x-wiki ==Week0 Preferences== Materials Used For Project 3: *Cat 5 cable *Vernier Photogate *Lab Quest Tablet <gallery> File:Vernier Photogate.jpg|Vernier Photogate File:Lab Quest Tablet.jpg|Lab Quest Tablet File:Cat3 Ethernet TP cable.jpg|Cat 5 cable </gallery> ==Week1 Narrative== <gallery> File:Cut Cat 5 Cable.jpg|you need to cut off one side of the cat 5 cable and leave the other side to be plug into the Vernier Photogate. File:Cut Cat 5 Cable 2.jpg|after cutting the cat 5 cable, you'll need to strip off each plastic coating to expose the copper wire. File:Cut Cat 5 Cable plug into Vernier Photogate..jpg|after making the cut, i pluged in the other end into the Vernier Photogate to get a visual idea of how it works. </gallery> YouTube Video showing off the Verier Photogate: http://www.youtube.com/watch?v=v8rgyOOBPPU&feature=youtu.be this is a video showing how Verier Photogate works with the Lab Quest tablet combine. the lab Quest takes reading in a form of a graph. when you bridge the gap between the sensor it plots points on lab quest that can easy be downloadable and analysis. ==Week2 Narrative== http://www.vernier.com/support/sensor-pinouts/ https://en.wikipedia.org/wiki/Category_5_cable http://www.vernier.com/engineering/arduino/digital-sensors/photogates/ ==Week3 Narrative== ==Week4 Narrative== 6q25iunqcknftlb6gd0i61jkcyscs3y 0 202457 2835506 2808023 2026-10-02T12:57:30Z Flickstar888 3106813 Edited the grammatical errors in this book chapter. 2835506 wikitext text/x-wiki {{title|Schadenfreude:<br>Why do we feel pleasure in the suffering of others?}} {{MECR3|1=http://prezi.com/mzqkmuvaf-ii/?utm_campaign=share&utm_medium=copy}} __TOC__ ==Overview== [[File:Schadenfreude.png|thumb|170px|''Figure 1''. An artificially generated image of a human facial expression consistent with schadenfreude: lip press → side-smirk → narrowed eyes]] Schadenfreude is a complex emotion that we feel when others suffer a misfortune (see Figure 1). However, instead of feelings of sympathy, schadenfreude evokes feelings of joy and pleasure. Schadenfreude has been considered immoral and malicious, and is closely linked to envy, one of the seven biblical sins (Takahashi, et al., 2009). For these reasons, many have argued that schadenfreude is harmful to social relations (Heider, 1958). Other research has attempted to combat that idea, as they argue that schadenfreude is a healthy emotion, despite the fact that it is not always appropriate or polite to share it with others (Spurgin, 2015). Understanding what schadenfreude is, where it comes form in terms of psychological theories, and why we encounter feelings of schadenfreude, will help to understand and improve on our emotional lives. == What is schadenfreude? == Schadenfreude is a German word which translates to the pleasure which is derived from the misfortune of others (Leach, Spears, Branscombe, & Doosje, 2003). Heider (1958) discussed how schadenfreude is a malicious emotion as it is an incongruous reaction to another person's misfortune. Heider (1958) is saying that instead of being sympathetic when another person is suffering, which could be considered the socially acceptable response, feelings of pleasure are seen as taboo and immoral (Leach, 2003). This feeling is typically seen as shameful or as a moral failing (Spurgin, 2015). Many people hide their feelings of schadenfreude, and many may not even realise that they are feeling pleasure at another person's misfortune. This can stem from things such as gloating or joy at your basketball team winning a game. Both have emotions of schadenfreude behind them. Schadenfreude has its roots in [[w:Social_comparison_theory|Social Comparison Theory]]. This theory, largely influenced by Festinger (1954), states that we evaluate our abilities and opinions by comparing our views with others, and that we want people in similar groups to like us, so will change our wants and beliefs to match theirs (Myers, 2014). Myers (2014) also describes social comparison as evaluating our abilities and opinions by comparing ourselves to others. As schadenfreude is a social comparison, where you are comparing yourself against the misfortune of someone else, you are forming an opinion or judging your own abilities on another person's misfortune. Schadenfreude is a complex cognitive emotion that has many different reasons as to why we feel it (Reeve, 2015). Schadenfreude can be derived from feelings of envy, instability in one's own self-worth, personal gain, when it is believed that the misfortune is deserved, along with biological factors. == Why do we feel pleasure in the suffering of others? == === The role of self-evaluation === When a person’s positive self-evaluation is threatened or harmed, they may have a strong motivation to protect or restore their self-evaluation (Van Dijk, 2013). One possible course of action to achieving this positive self-view can involve comparing one’s own situation to that of another person (Van Dijk, 2013). As a result, comparing another person’s misfortune may provide a sense of self-worth or value to one's own life. This means that people can use social comparisons and enjoy the misfortune of others as it provides a more positive self-evaluation. [[File:Eduardo Zamacois y Zabala - Regreso al convento.jpg |thumb|250px|''Figure 2''. Monks taking pleasure in the misfortune of another monk, by Eduardo Zamacois y Zabala]] Research conducted by van Dijk, Ouwerkerk, Wesseling, & van Koninbruggen (2011) supports the idea that schadenfreude can be intensified by a threat to our self evaluation. They hypothesise that another reason for people to feel schadenfreude is because it satisfies their need to view themselves positively (van Dijk, 2011) (see Figure 2). This is argued in [[w:Social_comparison_theory|Social Comparison Theory]] which suggests that events and experiences that satisfy our concerns elicit positive emotions, whereas threat or harm will produce negative emotions (van Dijk, 2011). A way that people can make themselves feel better, according to [[w:Social_comparison_theory|Social Comparison Theory]], is to compare themselves to those who are less fortunate, also called 'downward social comparison' (van Dijk, 2011). Therefore, it is possible to argue that those who are suffering from self-evaluation threat (and experiencing negative emotions), will use downward social comparison to help elicit positive emotions (Wills, 1981). The aim of Van Dijk's and his colleagues' (2011) research was to demonstrate that self-evaluation threat intensified schadenfreude in both threat-related and threat-unrelated domains. They were able to find that a threat to self-evaluation caused higher feelings of schadenfreude, and this was also possible to provoke in a threat-unrelated domain. This shows that self-evaluation can play a role towards feelings of schadenfreude. === Envy === [[File:Children marbles.jpg|thumb|225px|Figure 3. ''Envy shown in children with marbles'']] Envy (see Figure 3) has contradicting results when it came to schadenfreude. Many argued that there was a link between schadenfreude and envy, while others argued against this. van Dijk et al. (2006) investigated these contradictory results, and found that there is a link between schadenfreude and envy, but only when the misfortune fell upon someone who had some basis of similarity (e.g., gender). Their results found that if participants learnt about a misfortune of the opposite gender, schadenfreude would not be experienced (van Dijk, et al., 2006). However, when the same gender was identified as suffering misfortune, schadenfreude was identified. Smith, Powell, Combs, and Schurtz (2009) also show the correlation between envy and schadenfreude. They claim that envy is the polar opposite of a downward social comparison (Heider, 1958), however, when a misfortune occurs to someone who is envied, it transformed the comparison to a downwards one (Smith, et al., 2009). Conflicting reports on whether schadenfreude and envy are linked have been found, yet Smith et al., (2009), were able to replicate results where students who were enviable of another student felt greater schadenfreude when the person they envied suffered a misfortune, compared with those who were not in the envy group in the experiment. This provides empirical evidence that envy can lead to increased feelings of schadenfreude (Smith, et al., 2009). Smith et al., (2009) continue to remark that superiority to others does not always lead to envy, but when it does, this greatly increases the likelihood of schadenfreude. === In-Group Inferiority === An in-group refers to when an individual will recognise themselves as part of a group when they identify with them on some sort of level. For example, when someone identifies with a sporting group e.g. a football team, they begin to become part of the in-group. Another example is when people associate themselves with their university, an in-group forms. In-group inferiority refers to how people can feel pleasure at the misfortune of others in an in-group situation. For example, when a football team wins, that group will feel a sense of joy at the misfortune of the other team. Smith et al. (2009) suggests that when people identify with a group, the group becomes part of the individual, and the individual becomes part of the group. Leach et al. (2003) argue that schadenfreude is only evident when a third party or situation is the one that causes the misfortune, meaning that schadenfreude cannot occur if the pleasure is experienced when you are the cause of another persons misfortune. They suggest that schadenfreude should increase when an outer-group suffer misfortune in an area of high interest to the in-group members. They also delve into the idea that in-group inferiority will increase feelings of schadenfreude (Leach, et al., 2003; Leach and Spears, 2009). Leach et al. (2003) were successful in showing that when an individual felt more passionately about what formed the group (e.g. football) higher levels of schadenfreude were evoked when a third party suffered misfortune (e.g. lost a football match), and those who were less passionate, yet still were associated with the in-group had lesser feelings of schadenfreude. They were also able to demonstrate that schadenfreude was increased when feelings of in-group inferiority were experienced, however, this only affected those with lower interests (Leach, et al., 2003). Leach et al. (2003) also express that the threat to in-group inferiority and the increase in schadenfreude to those with higher interests was not seen, as those who had higher interests were already experiencing higher levels of schadenfreude. === Personal Gain (Competition) === Smith, et al. (2009) argue that the emotion of schadenfreude can be a result of a personal gain. They liken this to competition, where when you, or your team wins, you feel pleasure and this is ultimately in the suffering of the other team (Smith, et al., 2009). This idea of competition is seen in other aspects of life, and more often in day to day situations. It is arguably under-appreciated as to how often schadenfreude appears in a competitive everyday situation (Smith, et al., 2009). For example, if you are up for a new job, there is most likely going to be more than one person up for the position, and if you are successful in the process, you will most likely feel joy. This feeling of schadenfreude is one that is less ugly compared to other feelings derived from other places such as envy. A competitive nature is somewhat highly regarded (as seen with our tendency to highlight sports, and sports people), and seen to be quite natural (Smith, et al., 2009). Smith et al., (2009) discuss how this idea of personal gain is also evident in politics. Combs, Powell, Schurtz, and Smith (2009), conducted an experiment in the United States where they assessed whether schadenfreude was felt with political associations. They tested this by primarily assessing the political identification of students, then by asking them to read an article which made out something embarrassing (or unfortunate) about the party leaders for both their party and the opposing party (Combs, et al., 2009). They found that schadenfreude was present when participants were shown articles about the opposing parties, and that the level of schadenfreude found depended on how affiliated one was with their own political party (Combs, et al., 2009). This also ties in with the idea of in-group identification, as these examples of schadenfreude are mostly group based successes or failures. They still hold the idea that when your group wins, you feel pleasure - at the misfortune of others. These findings emphasize the fact that schadenfreude is much more common than we would like to admit, is found in everyday life, and it is often regarded as natural and praised (Smith, et al., 2009). === Deserved Misfortune === Another justification for schadenfreude is the sense of deserved misfortune. When we feel that the misfortune that one has suffered is deserved, a feeling of pleasure is derived. It is argued that the feeling of deserved misfortune, which creates the feeling of schadenfreude, is a form of karmic retribution and gives us a sense of equilibrium (Lerner, & Miller, 1978). van Dijk, Ouwerkerk, Goslinga, & Nieweg (2005) showed the first empirical evidence on deserved misfortune and its link to schadenfreude. They showed evidence of schadenfreude increasing when it was perceived that the misfortune was more deserved (van Dijk, et al., 2005). They used a manipulation of responsibility to obtain differences in deserved misfortune, which led to the evidence that schadenfreude and a feeling of deserved misfortune are linked (van Dijk et al., 2005). Smith et al., (2009) discuss how this is also linked with hypocrisy. They explain that when we feel someone has been a hypocrite, we feel pleasure in the form of schadenfreude, at their misfortune. This is because we believe that the misfortune they are suffering is deserved. Smith, et al., (2009) yielded results in an experiment to examine the links between schadenfreude, deserved misfortune, and hypocrisy. They asked participants to read an article that presented an interview with a fellow student, where the student was either part of a campus organization about increasing academic integrity (high hypocrisy) or a student who was part of a French club (low hypocrisy) (Smith,et al., 2009). Participants were then shown a second article which said that the fellow student (in either case) was caught for plagiarism (Smith, et al., 2009). The results showed that those who were in the high hypocrisy group showed higher feelings of schadenfreude and that the student deserved the misfortune in comparison to those who were in the low hypocrisy group (Smith, et al. 2009). Smith, et al. (2009) also found similar outcomes when they changed the first article to be the same for all participants, and the manipulation came in the second article, where the other student was either caught in an immoral action that either matched the initial action that they were fighting against, or something completely unrelated. The results showed that when the immoral action matched that of the initial action higher levels of deserved misfortune and schadenfreude were felt. Unfortunately, this exact study was never published on its own, which questions whether there were problems with integrity in the research. Pietraszkiewicz (2013) discusses how schadenfreude and deserved misfortune are correlated to a just world belief. It was found that a threat to one's world belief increased one's pleasure at another's' misfortune. Pietraszkiewicz (2013) argues that when failure is deserved, the greater the responsibility of the failure is, therefore, more schadenfreude is felt. === Biological components === ==== The role of Oxytocin ==== Research that was conducted by Shamay-Tsoory, et al. (2009) investigated the role of oxytocin in envy and gloating, which are both related to schadenfreude. Shamay-Tsoory, et al. (2009) discuss how oxytocin (a peptide hormone) has been shown to have implications in the social behaviour of humans and mammals. Many of the research into oxytocin looks at maternal behaviours such as contraction regulation in labour, as well as parental behaviours like trusting collaborators. They suggest that previous research has shown that oxytocin release is related to pro-social behaviours (Sharmay-Tsoory, et al., 2009). Seeing as pro-social behaviours are increased by oxytocin, your negative social behaviours, like envy and gloating, would logically be reduced. However, there has been an indication that this is not the case (Sharmay-Tsoory, et al., 2009). Sharmay-Tsoory et al. (2009) conducted an experiment which looked at the increase of oxytocin levels and its effect on these negatively perceived behaviours, such as schadenfreude. They concluded that gloating and envy, or schadenfreude, showed significantly higher rates of these emotions when oxytocin was given (Sharmay-Tsoory, et al., 2009). This research has provided evidence that oxytocin increases varying behaviours which are related to social behaviour, which in many roles are associated with parenting. ==== Neural Correlates ==== [[File:MRI anterior cingulate.png|alt=|thumb|Figure 4. ''MRI of anterior cingulate cortex'' ]] [[File:Dopamine Pathways.png|alt=|thumb|Figure 5. ''Position of the Ventral Striatum'']] Envy and schadenfreude are related emotions. Takahashi, et al. (2009) looked at the areas of the brain that were active when feelings of envy and schadenfreude were evoked. Using functional magnetic resonance imaging (fMRI) researchers looked for activity in the dorsal anterior cingulate cortex (dACC) (seen in image 2.) when envy was felt, as the anterior cingulate cortex is the area that is activated when our positive self-concept is being conflicted with external information, social pain, or cognitive conflicts (Takahashi, et al., 2009). When investigating the emotion of schadenfreude they were looking for activation in the ventral striatum, which is the central node of the rewards processing area (Takahashi, et al., 2009). The reward would be the joy that is derived in schadenfreude. Takahashi, et al. (2009) found that both areas which were targeted in their respective trials activated when the respective emotion was emitted. They found that when people had higher levels of schadenfreude, greater activation was seen in the ventral striatum. This was also found to be the case when investigating envy. Greater levels of envy showed higher activation in the dACC (Takahashi, et al., 2009). == Activities == ; Video To see an example of Pleasure derived from others misfortune follow this link to YouTube [https://www.youtube.com/watch?v=mcRyTdFKjPU 14 awesome viral video fails in 30 seconds] ; Poll After watching this video please complete this short poll to see how others feel when it comes to Schadenfreude [http://www.easypolls.net/poll.html?p=562c7466e4b09c75340b5249 Link to poll] Why do some people find videos like this funny? They could be experiencing feelings of schadenfreude, as they are getting pleasure from the suffering of others. But why do we feel this way? == Quiz == <quiz display="simple"> {What is Schadenfeude? |type="()"} - Freud's son + Pleasure derived from others misfortune - Pleasure derived from others fortune - Pain derived from others misfortune {What Peptide Hormione plays a role in Schadenfreude? |type="()"} - Oxycontin - glucocorticoids - Prolactin + Oxytocin {Which of the following does NOT play a role in Schadenfreude? |type="()"} - Envy - Self-worth + Anhedonia - Deserved misfortune {Which area of the brain is stimulated when you feel the emotion of schadenfreude? |type="()"} - dorsal Anterior Cingulate Cortex + Ventral Striatum - Prefrontal Cortex - Subgenual Cingulate </quiz> ==Conclusion == Schadenfreude is a complex emotion which can have many levels (van Dijk, 2011). It has many underlying concepts which relate to Social Comparison Theory, especially when evaluating the role of self-evaluation and schadenfreude. It is more commonly seen as a morally disturbed emotion, especially when feelings of envy, self-evaluation, and in-group inferiority are causes. However, it is seen quite commonly in everyday life, especially when it comes to situations or individuals with a competitive nature. There are many layers that underlie why schadenfreude occurs. Emotions such as envy, feelings of deservingness, personal gain, in-group inferiority and self-evaluation can all play a role in schadenfreude and why we feel this emotion, and there can be more than one reason as to why we experience schadenfreude. Other biological reasons, such as the role of oxytocin, and activity in the ventral striatum also play a role in feelings of schadenfreude. Schadenfreude is difficult to elicit in a clinical setting in an ethical way, which may be why conflicting results have been obtained for much of the research. Limitations in assessing schadenfreude may include social biases, as schadenfreude is an emotion that is generally considered immoral. This can lead to participants under-reporting their feelings of schadenfreude when being asked about it. A possible solution to this would be to conduct the schadenfreude eliciting part of the experiment under fMRI, as activity in the ventral striatum has been linked to schadenfreude, and it may be another way to see if someone is experiencing schadenfreude, without self-reporting. This may become more costly and less time effective, which would be a reason to stay clear of this research technique, however, it does become an option. Due to the many aspects and layers of schadenfreude, van Dijk et al., (2011) suggest that future research into which determinants effect schadenfreude under what circumstances will aid in further understanding why we feel pleasure in others misfortune. There may possibly be other reasons as to why we feel schadenfreude, greater research into the biological aspects of schadenfreude may also enhance our understanding of this emotion. Through investigating schadenfreude and the reasons why we might feel this emotion can aid to enrich our understanding and improve on out emotional lives, as when an experience of schadenfreude is likely to occur or is experienced, taking a step back and evaluating why we are feeling this emotion may lead to a greater emotional understanding and awareness. ==See also== * [[Motivation and emotion/Book/2013/Deservingness and emotion#Schadenfreude|Deservingness and emotion]] (Book chapter, 2013) * [[Motivation and emotion/Book/2011/Envy|Envy]] (Book chapter, 2011) * [[w:Schadenfreude|Schadenfreude]] (Wikipedia) ==References== {{Hanging indent|1= Combs, D. J. Y., Powell, C. A. J., Schurtz, D. R., & Smith, R. H. (2009). 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Group Influence. In Myer, D. G. (Eds.), ''Social Psychology'' (287). North Ride, N.S.W.: McGraw-Hill Pietraszkiewicz. A. (2013). Schdenfeude and just world belief. ''Australian Journal of Psychology, 65'', 188-194. doi: 10.1111/ajpy.12020 Reeve, J. (2015). ''Understanding motivation and emotion'' (6th ed.). Hoboken, NJ: Wiley. Shamay-Tsoory, S. G,. Fischer, M., Dvash, J., Harari, H., Perach-Bloom, N., & Levkovitz, Y. (2009). Intranasal administration of oxytocin increases envy and schadenfreude (gloating). ''Biological Psychiatry, 66(8)'', 864-870. doi: 10.1016/j.biopsych.2009.06.009 Smith, R. H., Powell, C. A. J., Combs, D. J. Y., & Schurtz. D. R. (2009). Exploring the when and why of schadenfreude. ''Social and Personality Psychology Compass, 3(4)'', 530-546. doi: 10.1111/j.1751-9004.2009.00181.x Spurgin, E. (2015). An emotional-freedom defense of schadenfreude. ''Ethical Theory and Modern Practice, 18'', 767-784. doi: 10.1007/s10677-014-9550-8 Van Dijk, W. W. (2013). Why do we sometimes enjoy the misfortune of others? ''The Inquisitive Mind'' Retrieved from: http://www.in-mind.org/blog/post/why-do-we-sometimes-enjoy-the-misfortune-of-others van Dijk, W. W., Goslinga, O. S., Nieweg, M., & Gallucci, M. (2006). When people fall from grace: reconsidering the role of envy in schadenfreude. ''Emotion, 6(1)'', 156-160. doi: 10.1037/1528-.3542.6.1.156 van Dijk, W. W., Ouwerkerk, J., Goslinga, S., & Nieweg, M. (2005). Deservingness and Schadenfreude. ''Cognition and Emotion, 19(6),'' 933-939. doi: 10.1080/02699930541000066 van Dijk, W. W., Ouwerkerk, J., Wesseling, Y. M., & van Koningsbruggen, G. M. (2011). Towards understanding pleasure at the misfortunes of others: the impact of self-evaluation threat on schadenfreude. ''Cognition and Emotion'', 25(2), 360-368. doi: 10.1080/02699931.2010.487365 Wills, T. A. (1981). Downward comparison principles in social psychology. ''Psychological Bulletin, 90(2),'' 245-271. }} ==External links== * [http://www.livescience.com/17398-schadenfreude-affirmation.html Schadenfreude Explained: Why We Secretly Smile When Others Fail] (livescience.com) * [http://www.wsj.com/articles/schadenfreude-is-in-the-zeitgeist-but-is-there-an-opposite-term-1434129186 Schadenfreude Is in the Zeitgeist, but Is There an Opposite Term?] (Wall Street Journal) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Schadenfreude]] tj0hhtdc0wi4gh9qwn8wlm53wsulv9b 0 203942 2835562 2834562 2026-10-02T20:24:48Z Young1lim 21186 /* Lambda Calculus */ 2835562 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20260915.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260928.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] q915ae5m2yusc8ogd74o1xtww4pfnxj 2835564 2835562 2026-10-02T20:25:59Z Young1lim 21186 /* Lambda Calculus */ 2835564 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20260915.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260929.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] 8g20jz9e0snhmzes8sy5lqv9twkk0ns 0 235131 2835576 2835224 2026-10-02T21:41:11Z The Citer 3110681 /* Xenacoelomorpha */ 2835576 wikitext text/x-wiki [[Image:Animalia diversity.jpg|thumb|300px|right|Some examples of animal diversity.]] A Phylum (pl. Phyla) is the largest formal major grouping within animal taxonomy. This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement. This list is also available in [[/Table/|table form]]. Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Animal Phyla. Other sources may combine or split these listed Phyla. However, at this time, the list presented here should stand in good stead for an introduction to the topic of animal diversity. There are approximately 1.9 million animal species that have been described by science. This list tries to give the following information on each Phylum: *Phyllum Name *A link to a sub-page discussing that Phylum in more detail (if it yet exists) *Name Meaning (in English) *An English Common Name, where one is in regular use *Distinguishing characteristics of animals within the Phylum *An approximate number of species described within that Phylum. Since zoology fortunately refuses to stand still, this number can change. Oh! And there's a quiz at the end. See also [[/Lesson Plan for Animal Phyla/]]. Or head to [[Introduction to Taxonomy]] for more on that topic. You can also visit [[Plant Divisions (Phyla)]], the equivalent page for plants. ==Acanthocephalae== [[Image:Macracanthorhynchus_hirudinaceus_adult_BAM1.jpg|thumb|100px|left|An Acanthocephalan, ''Macracanthorhynchus hirudinaceus'', adult.]] [[/Acanthocephala/]] Name Meaning: horny head English Common Name: Horny-headed worms Major distinguishing characteristics: Reversible spiny proboscis Approximate number of species described: 1,151 ==Acoelomorpha== [[File:Acoelomorpha.jpg|thumb|100px|left|Acoelomorpha]] [[/Acoelomorpha/]] Name Meaning: Without gut English Common Name: Acoels Major distinguishing characteristics: No mouth or alimentary canal Approximate number of species described: 50,000 == Annelida == <div id="Annelida"></div> [[Image:Reef0200.jpg|thumb|100px|left|An feather duster worm of the family Sabellidae in parchment tube.]] [[/Annelida/]] Name Meaning: Little ring English Common Name: Segmented worms, annelids Major distinguishing characteristics: Multiple circular segments Approximate number of species described: 22,000 modern ==Arthropoda== [[Image:Chelicerae_%26_pedipalps_%288689230685%29.jpg|thumb|100px|left|A scorpion, an example of an Arthropod]] [[/Arthropoda/]] Name Meaning: Jointed foot English Common Name: Arthropods Major distinguishing characteristics: Chitin exoskeleton Approximate number of species described: 1,134,000+ ==Brachiopoda== [[Image:Isocrania_costata_Sowerby_1823.jpg|thumb|100px|left|Brachiopod ''Isocrania costata''.]] [[/Brachiopoda/]] Name Meaning: Arm foot English Common Name: Lamp shells, brachiopod Major distinguishing characteristics: Lophophore and pedicle Approximate number of species described: between 300 and 500 extant ==Bryozoa== [[Image:Freshwater_Bryozoan234.JPG|thumb|100px|left|A freshwater Bryozoan, species unknown.]] [[/Bryozoa/]] Name Meaning: Moss animals English Common Name: Moss animals, sea mats, bryozoans Major distinguishing characteristics: Lophophore, no pedicle, ciliated tentacles Approximate number of species described: about 5,000 living species ==Chaetognatha== [[Image:Chaetognatha.PNG|thumb|100px|left|Chaetognathans.]] [[/Chaetognatha/]] Name Meaning: Longhair jaw English Common Name: Arrow worms Major distinguishing characteristics: Chitinous spines either side of head, fins Approximate number of species described: about 100 modern species ==Chordata== [[Image:Clavelina lepadiformis (Müller, 1776) - Banyuls-sur-Mer - 04.83.jpg|thumb|100px|left|''Clavelina lepadiformis'', a non-vertebrate chordates.]] [[/Chordata/]] Name Meaning: Cord English Common Name: Chordates Major distinguishing characteristics: Hollow dorsal nervous chord Approximate number of species described: about 100,000+ ==Cnidaria== [[Image:Cynarina_lacrymalis.jpeg|thumb|100px|left|''Cynarina lacrymalis'' an anthozoan member of the Phylum Cnidaria.]] [[/Cnidaria/]] Name Meaning: Stinging nettle English Common Name: Coelenterates, cnidarians, sea anemones, jellies, hydra Major distinguishing characteristics: Cnidocytes (stinging cells) Approximate number of species described: about 11,000 ==Ctenophora== [[Image:LightRefractsOf comb-rows of ctenophore Mertensia ovum.jpg|thumb|100px|left|''Mertensia'', a Ctenophore.]] [[/Ctenophora/]] Name Meaning: Comb bearer English Common Name: Comb jellies, Ctenophores Major distinguishing characteristics: Eight "comb rows" of fused cilia Approximate number of species described: about 200 modern species ==Cycliophora== [[Image:Cycliophora - Symbion pandora.png|thumb|100px|left|''Symbion pandora'', a Cycliphoran.]] [[/Cycliophora/]] Name Meaning: Wheel carrying English Common Name: Symbion Major distinguishing characteristics: Circular mouth surrounded by small cilia Approximate number of species described: at least 3 ==Echinodermata== [[Image:Crinoideos.jpg|thumb|100px|left|''Crinoideos'', an example of Ecinoderms.]] [[/Echinodermata/]] Name Meaning: Spiny skin(Body) English Common Name: Echinoderms Major distinguishing characteristics: Five-fold radial symmetry, mesodermal layer, calcified spines Approximate number of species described: about 7,000 living species and 13,000 extinct ones ==Entoprocta== [[Image:Barentsa discreta suzukokemusi01.JPG|thumb|100px|left|''Barentsa discreta suzukokemusi'', a species of goblet worm.]] [[/Entoprocta/]] Name Meaning: Inside anus English Common Name: Goblet worm Major distinguishing characteristics: Anus inside ring of cilia Approximate number of species described: about 150 ==Gastrotricha== [[Image:Chaetonotus (Chaetonotus) polyspinosus.jpg|thumb|100px|left|''Chaetonotus polyspinosus'' a species of meiofauna.]] [[/Gastrotricha/]] Name Meaning: Hair stomach English Common Name: Hairybellies, hairybacks Major Distinguishing Characteristics: Two terminal adhesive tubes Approximate number of species described: about 690 ==Gnathostomulida== [[Image:Gnathostomula paradoxa Sylt.tif|thumb|100px|left|''Gnathostomula paradoxa'', a jaw worm.]] [[/Gnathostomulida/]] Name Meaning: Jaw orifice English Common Name: Jaw worms Major distinguishing characteristics: Approximate number of species described: about 100 ==Hemichordata== [[Image:Glossograptus whitfieldii 01.jpg|thumb|100px|left|Glossograptus whitfieldii, a fossil hemichordate.]] [[/Hemichordata/]] Name Meaning: Half cord English Common Name: Acorn worms Major distinguishing characteristics: Stomochord in collar Approximate number of species described: about 100 living species ==Kinorhyncha== [[Image:Paracentrophyes quadridentatus.jpg|thumb|100px|left|''Paracentrophyes quadridentatus'', a mud dragon.]] [[/Kinorhyncha/]] Name Meaning: Motion snout English Common Name: Mud dragons Major distinguishing characteristics: Eleven segments, each with a dorsal plate Approximate number of species described: about 150 ==Loricifera== [[Image:Pliciloricus enigmatus.jpg|thumb|100px|left|A member of the Phylum Loricifera, ''Pliciloricus enigmatus''.]] [[/Loricifera/]] Name Meaning: Corset bearer English Common Name: Loriciferans Major distinguishing characteristics: Umbrella-like scales at each end Approximate number of species described: about 122 ==Micrognathozoa== [[Image:Limnonathia, drawing.tif|thumb|100px|left|''Limnonathia'', drawing, a Micrognathozoan.]] [[/Micrognathozoa/]] (sometimes called Phylum Gnathifera) Name Meaning: Tiny jaw animals English Common Name: Micrognathozoans Major distinguishing characteristics: Accordion like extensible thorax Approximate number of species described: 1 ==Mollusca== [[Image:Shells01.jpg|thumb|100px|left|A group of Gastropod shells, Phylum Mollusca.]] [[/Mollusca/]] Name Meaning: Thin shell English Common Name: Mollusks or molluscs Major distinguishing characteristics: Muscular foot and Mantle, round shell Approximate number of species described: 85,000 ==Nematoda== [[Image:Mermis nigrescens beentree.jpg|thumb|100px|left|''Mermis nigrescens beentree'', a nematode worm.]] [[/Nematoda/]] Name Meaning: Thread like English Common Name: Round worms, Nematodes Major distinguishing characteristics: Round cross section, keratin, cuticle Approximate number of species described: 80 000 - 1 million ==Nematomorpha== [[Image:Gordiidae A MRKVICKA.JPG|thumb|100px|left|A Gordian worm, Phylum Nematomorpha.]] [[/Nematomorpha/]] Name Meaning: Thread form English Common Name: Horsehair worms Major distinguishing characteristics: Approximate number of species described: about 320 ==Nemertea== [[Image:Micrura alaskensis.png|thumb|100px|left|Micrura alaskensis, a ribbon worm.]] [[/Nemertea/]] Name Meaning: A sea nymph English Common Name: Ribbon worms Major distinguishing characteristics: Approximate number of species described: about 1200 ==Onychophora== [[Image:Euperipatoides kanangrensis crop.jpg|thumb|100px|left|''Euperipatoides kanangrensis'', a velvet worm.]] [[/Onychophora/]] Name Meaning: Claw bearer English Common Name: Velvet worms Major distinguishing characteristics: Legs tipped by chitinous claws Approximate number of species described: about 200 modern ==Orthonectida== <div id="Orthonectida"></div> [[Image:Orthonetida dict flat and cylinder.png|thumb|100px|left|Orthonetida, an Orthonectidan.]] [[/Orthonectida/]] Name Meaning: Straight swim English Common Name: Orthonectida Major distinguishing characteristics: Approximate number of species described: about 20 ==Phoronida== [[Image:Phoronis Maria Grazia Montanucci2.jpg|thumb|100px|left|'' Phoronis hippocrepis'', a species of horseshoe worm.]] [[/Phoronida/]] Name Meaning: Zeus' mistress English Common Name: Horseshoe worms Major distinguishing characteristics: U-shaped gut Approximate number of species described: 20 ==Placozoa== [[Image:Trichoplax adhaerens photograph.png|thumb|100px|left|''Trichoplax adhaerens'', the only Placozoan.]] [[/Placozoa/]] Name Meaning: Plate animals English Common Name: none Major distinguishing characteristics: Approximate number of species described: 1 ==Platyhelminthes== [[Image:Gigantolina.elongata.in.vivo.png|thumb|100px|left|''Gigantolina elongata'', a flat worm.]] [[/Platyhelminthes/]] Name Meaning: Flat worms English Common Name: Flat worms Major distinguishing characteristics: Approximate number of species described: about 25,000 ==Porifera== [[Image:Eponge à déterminer (1).jpg|thumb|100px|left|An unidentified sponge.]] [[/Porifera/]] Name Meaning: Pore bearer English Common Name: Sponges Major distinguishing characteristics: Perforated interior wall Approximate number of species described: over 5,000 modern ==Priapulida== [[Image:Adult priapulid.jpg|thumb|100px|left|''Priapulus caudatus'', a Priapulid worm.]] [[/Priapulida/]] Name Meaning: Penis English Common Name: Priapulid worms Major distinguishing characteristics: Retractable proboscis surrounded by papillae Approximate number of species described: 17 ==Rhombozoa== <div id="Rhombozoa"></div> [[Image:Dicyema japonicum.png |thumb|100px|left|''Dicyema japonicum'', a rhombozoan.]] [[/Rhombozoa/]] Name Meaning: Rhombus animal English Common Name: Rhombozoans Major distinguishing characteristics: Single axial cell surrounded by ciliated cells Approximate number of species described: 75 ==Rotifera== [[Image:20090730 020239 Rotifer.jpg|thumb|100px|left|An unidentified rotifer.]] [[/Rotifera/]] Name Meaning: Wheel bearer English Common Name: Rotifers Major distinguishing characteristics: Anterior crown of cilia Approximate number of species described: about 2000 ==Tardigrada== [[Image:SEM image of Milnesium tardigradum in active state - journal.pone.0045682.g001-2.png|thumb|100px|left|Tardigrade ''Milnesium tardigradum'' in active state.]] [[/Tardigrada/]] Name Meaning: Slow step English Common Name: Water bears Major distinguishing characteristics: Four segmented body and head Approximate number of species described: 1,000+ ==Xenacoelomorpha== [[Image:Xenoturbella bockii longitudinal section.svg|thumb|100px|left|''Xenoturbella bockii'' longitudinal section.]] [[/Xenacoelomorpha/]] Name Meaning: Strange hollow form English Common Name: none Major distinguishing characteristics: Ciliated deuterostome Approximate number of species described: 435 ==Quizzes== * [[/Quiz/|Animal Phyla Quiz]] ==References== {{reflist}} *Brusca, Richard C., Wendy Moore, Stephen M. Shuster. Invertebrate Zoology 3rd ed. 2016. Sinauer. 1104 pp. *[https://en.wikipedia.org/wiki/Phylum Phylum] on Wikipedia [[Category:Lists]] [[Category:Taxonomy]] [[Category:Zoology]] [[Category:Animals]] {{science}} {{biology}} ddlqbnbkfcbfqvfj1igh5cpkwr7f3rr 0 235142 2835551 2697689 2026-10-02T18:47:33Z The Citer 3110681 Added phylogeny! 2835551 wikitext text/x-wiki [[Image:Macracanthorhynchus_hirudinaceus_adult_BAM1.jpg|thumb|300px|right|An Acanthocephalan, ''Macracanthorhynchus hirudinaceus'', adult.]] *Acanthocephala *Name Meaning: Thorny head *English Common Name: Thorny-headed worms *Major distinguishing characteristics: Reversible spiny proboscis *Approximate number of species described: 1,420 ==Natural History== All known '''Acanthocephalans''' are intestinal '''parasites''' of '''vertebrates'''. Some spend their early development as parasites of '''invertebrates''', but must move to a vertebrate '''host''' in order to '''reproduce'''. They are most commonly found in '''fishes''', but may be found in any type of vertebrate. They lodge by their "horny" (spine covered) '''proboscis''' into the intestinal wall of their host. They '''ingest''' directly through their body wall. ==Taxonomy== There are over '''1,400 species''' described, in '''four Classes'''. The four Classes are *Archiacanthocephala *Eoacanthocephala *Palaeacanthocephala *Polyacanthocephala ==Phylogeny== {{clade |label1=Syndermata |1={{clade |1=Seisonida |2='''Acanthocephala''' }} |label2=Eurotatoria |2={{clade |1=Bdelloidea |2=Monogononta }} }} ==Anatomy== Acanthocephala are a simple '''tubular worm''' with a spiny or "thorny" proboscis, which they use to embed themselves into the intestinal wall of their host. ==The Fossil Record== While no adult specimens of acanthocephalans have been found in the fossil record, their eggs have been recorded as far back as the Cretaceous. ==Quiz== * [[/Acanthocephala Quiz/]] ==References and Further Reading== *[http://eol.org/pages/2/overview Acanthocephala] at Encyclopedia of Life *[http://tolweb.org/Acanthocephala/20452/2006.07.28 Acanthocephala] at Tree of Life *[https://www.ncbi.nlm.nih.gov/pubmed/24261131 Classification of the Acanthocephala] *Labandeir, Conrad C. Paleobiology of Predators, Parasitoids, and Parasites: Death and Accomodation in the Fossil Record of Continental Invertebrates. ''in'' Kowalewski, M., and P.H. Kelley, (eds.), 2002. The Fossil Record of Prédation. Paleontological Society Papers, 8: 211-249. Available online at [https://repository.si.edu/bitstream/handle/10088/5959/Paleont_Soc_Papers_2002a.pdf The Smithsonian Institution] *https://www.scielo.br/j/aabc/a/7YbvGFfkzr7dvFVwvYL9g8B/?lang=en {{Animalia}} [[Category:Animals]] [[Category:Taxonomy]] [[Category:Biology]] tqnzh957w1ezwbybicko7wb3odbe3vq 2835553 2835551 2026-10-02T18:48:59Z The Citer 3110681 2835553 wikitext text/x-wiki [[Image:Macracanthorhynchus_hirudinaceus_adult_BAM1.jpg|thumb|300px|right|An Acanthocephalan, ''Macracanthorhynchus hirudinaceus'', adult.]] *Acanthocephala *Name Meaning: Thorny head *English Common Name: Thorny-headed worms *Major distinguishing characteristics: Reversible spiny proboscis *Approximate number of species described: 1,420 ==Natural History== All known '''Acanthocephalans''' are intestinal '''parasites''' of '''vertebrates'''. Some spend their early development as parasites of '''invertebrates''', but must move to a vertebrate '''host''' in order to '''reproduce'''. They are most commonly found in '''fishes''', but may be found in any type of vertebrate. They lodge by their "horny" (spine covered) '''proboscis''' into the intestinal wall of their host. They '''ingest''' directly through their body wall. ==Taxonomy== There are over '''1,400 species''' described, in '''four Classes'''. The four Classes are *Archiacanthocephala *Eoacanthocephala *Palaeacanthocephala *Polyacanthocephala ==Phylogeny== {{clade |label1=Syndermata |1={{clade |1=Seisonida |2='''Acanthocephala''' }} |label2=Eurotatoria |2={{clade |1=Bdelloidea |2=Monogononta }} }} ==Anatomy== Acanthocephala are a simple '''tubular worm''' with a spiny or "thorny" proboscis, which they use to embed themselves into the intestinal wall of their host. ==The Fossil Record== While no adult specimens of acanthocephalans have been found in the fossil record, their eggs have been recorded as far back as the Cretaceous. ==Quiz== * [[/Acanthocephala Quiz/]] ==References and Further Reading== *[http://eol.org/pages/2/overview Acanthocephala] at Encyclopedia of Life *[http://tolweb.org/Acanthocephala/20452/2006.07.28 Acanthocephala] at Tree of Life *[https://www.ncbi.nlm.nih.gov/pubmed/24261131 Classification of the Acanthocephala] *Labandeir, Conrad C. Paleobiology of Predators, Parasitoids, and Parasites: Death and Accomodation in the Fossil Record of Continental Invertebrates. ''in'' Kowalewski, M., and P.H. Kelley, (eds.), 2002. The Fossil Record of Prédation. Paleontological Society Papers, 8: 211-249. Available online at [https://repository.si.edu/bitstream/handle/10088/5959/Paleont_Soc_Papers_2002a.pdf The Smithsonian Institution] *https://www.scielo.br/j/aabc/a/7YbvGFfkzr7dvFVwvYL9g8B/?lang=en {{Animalia}} [[Category:Animals]] [[Category:Taxonomy]] [[Category:Biology]] [[Category:Zoology]] toul0icmjqwtbdyejdl3p8nwt6fg7gq 0 235146 2835579 2697687 2026-10-02T21:45:05Z The Citer 3110681 Added phylogeny! 2835579 wikitext text/x-wiki [[File:Acoelomorpha.jpg|thumb|300px|right|Acoelomorpha]] *Acoelomorpha *Name Meaning: With gut *English Common Name: Acoels *Major distinguishing characteristics: No mouth or alimentary canal *Approximate number of species described: unknown ==Natural History== All but two freshwater species are marine. All are aquatic. They live between the grains of sand, in the water column as plankton, or crawling on the substrate or other organisms. ==Taxonomy== Some consider this group to be a Subphylum of Xenacoelomorpha. There are two Classes that traditionally belong to the Phylum regardless of their placement. *Acoela *Nemertodermatida ==Phylogeny== {{Clade |label1=Acoelomorpha |sublabel1=(may be a phylum or a subphylum) |1={{clade |label1=Acoela |1={{clade |label1= |state1=dashed |1=Actinoposthiidae [[File:Pseudactinoposthia parva.gif|60px]] |state2=dashed |2=Antigonariidae [[File:Antigonaria arenaria.jpg|60px]] |state3=dashed |3=Antroposthiidae [[File:Antroposthia axi.jpg|60px]] |4=Diopisthoporidae |state5=dashed |5=Nadinidae |state6=dashed |6=Paratomellidae |state7=dashed |7=Taurididae |8={{clade |label1= |1=Paratomellidae |label2=Bursalia |2={{clade |label1=Prosopharyngida |1={{clade |label1= |1=Hallangiidae |2={{clade |label1= |1=Hofsteniidae |2=Solenofilomorphidae }} }} |label2=Crucimusculata |2={{clade |label1= |1=Proporidae [[File:Proporus sp.png|60px]] |2={{clade |label1= |1=Isodiametridae [[File:Isodiametra pulchra.jpg|60x60px]] |2={{clade |label1= |1=Dakuidae |label2=Aberrantospermata |2={{clade |label1= |1=Convolutidae [[File:Convolutriloba retrogemma underside on aquarium glass Length 3mm.jpg|60px]] |2=Mecynostomidae }} }} }} }} }} }} }} |label2=Nemertodermatida |2={{clade |label1=Ascopariidae |1={{clade |label1= |1=''Ascoparia'' |2=''Flagellophora'' [[File:Flagellophora sp.png|60px]] }} |label2=Nemertodermatidae |2={{clade |label1= |1=''Nemertoderma'' |state2=dashed |2=''Nemertinoides'' [[File:Nemertinoides elongatus.png|60x60px]] |3={{clade |label1= |1=''Meara stichopi'' [[File:Meara stichopi.jpg|60px]] |2=''Sterreria'' [[File:Sterreria variabilis.png|60x60px]] }} }} }} }} }} ==Anatomy== The Acoelomorpha resemble flat worms. Their internal structures are so simple they have no real internal structures. ==The Fossil Record== There is only one known fossil of an acoelomorph, [[w:Gutticorpus|Gutticorpus]]. ==Quiz== * [[/Acoelomorpha Quiz/]] ==References and Further Reading== *[http://eol.org/pages/3028491/hierarchy_entries/52569368/overview Acoelomorpha] on Encyclopedia of Life *[https://en.wikipedia.org/wiki/Acoelomorpha Acoelomorpha] at Wikipedia {{Animalia}} [[Category:Animals]] [[Category:Taxonomy]] [[Category:Zoology]] [[Category:Biology]] bbl5q0v2v96475tau2kgupoyj806qfp 0 235160 2835580 2010563 2026-10-02T21:58:18Z The Citer 3110681 Added phylogeny! 2835580 wikitext text/x-wiki [[Image:Reef0200.jpg|thumb|300px|Right|An feather duster worm of the family Sabellidae in parchment tube.]] Annelida *Name Meaning: Little ring *English Common Name: Segmented worms *Major distinguishing characteristics: Multiple circular segments *Approximate number of species described: over 22,000 living ==Natural History== The life history of the Annelida is '''diverse'''. They live in '''terrestrial''' and '''aquatic''' '''environments''', including around '''deep sea hydrothermal vents'''. They are common, yet rarely seen because most Annelids are '''burrowing'''. Some are highly mobile, some seldom move once established. Some, such as '''earthworms''', are '''omnivorous''', eating any '''organic''' matter in their path. Others, such as the '''leaches''', are specialist in what they will eat. Leaches, for example, are '''specialist blood eaters'''. In all environments where they occur, Annelids are very important in '''moving and mixing the soil''', and breaking up and '''distributing organic material'''. This distribution helps plants get better access to '''nutrients'''. ==Taxonomy== The Phylum Annelida contains 7 Classes, 6 of which have living representatives: *Polychaeta *Clitellata *Oligochaeta (earthworms) *Branchiobdellida *Hirudinea (leeches) *Echiura *†Machaeridia *Echiura is a former Phylum that has been demoted to Class rank within the Annelida. ==Phylogeny== {{clade|style=font-size:80%;|label1='''Annelida''' |1={{clade |1={{clade|label1=Palaeoannelida  |1={{clade |1=Oweniidae |2=Magelonidae }}}} |2={{clade |1={{clade|label1=Chaetopteriformia  |1={{clade |1=''Apistobranchus'' |2={{clade |1=Psammodrilidae |2=Chaetopteridae }} }}}} |2={{clade |1={{clade |1=Amphinomida |2={{clade |1=''Lobatocerebrum'' |2=[[Sipuncula]] (May or not be a phylum on its own) }} }} |2={{clade|label1=Pleistoannelida |1={{clade |label1=Errantia|1={{clade |label2=Protodriliformia|2={{clade |1=Polygordiidae |2=Protodrilida }} |label1=Aciculata|1={{clade |1=Eunicida |2=Phyllodocida }} }} |label2=''?''|2=Myzostomida|state2=dashed |label3=Sedentaria|3={{clade |1=Orbiniida |2={{clade |1={{clade |1={{clade|1=Cirratuliformia|2=Siboglinidae (pogonophorans)}} |2={{clade|1=Sabellida|2=Spionida}} }} |2={{clade |1={{clade |1=Opheliida|label2=Capitellida|2={{clade|1=Capitellidae|2=Echiura}}}} |2={{clade |1={{clade|1=Terebelliformia|2=Maldanomorpha}} |2={{nowrap|Clitellata}}}} }}}} }} }} }} }} }} }} }} ==Anatomy== Annelids have long bodies. They are generally '''segmented''', the segments usually being visible '''externally'''. Each segment generally contains a full set of '''organs'''. The segments share a common '''digestive system''', '''circulatory system''', and '''nervous system'''. Three segments, the '''prostomium''' (1st segment), '''peristomium,''' (2nd segment), and '''pygidium''' (last segment), are '''specialized'''. The prostomium contains the '''brain''' and '''sense organs'''. The peristomium contains the '''mouth'''. The pygidium is the location of the '''anus'''. Annelids grow from behind. The '''growth region''' is just in front of the pygidium. The peristomium is the oldest segment. ==The Fossil Record== Annelids are '''soft bodied''' and so fossils are rare. The first Annelids probably evolved during the '''Precambrian''', before 541 million years ago. They are present in the '''Burgess Shale''' (about 505 million years old), though there may be some '''Australian''' fossil annelids about 518 million years old. There is a better fossil record of Annelids by 65 million years ago. ==Quiz== * [[/Annelida Quiz/]] ==References and Further Reading== *[https://en.wikipedia.org/wiki/Annelid Annelid] on Wikipedia *[http://animaldiversity.org/accounts/Annelida/ Annelida] on the Animal Diversity Web, University of Michigan *[http://www.ucmp.berkeley.edu/annelida/annelida.html Introduction of the Annelida] at the University of California Museum of Paleontology *[http://eol.org/pages/36/overview Annelida] on the Encyclopedia of Life [[Category:Animals]] [[Category:Taxonomy]] [[Category:Zoology]] [[Category:Biology]] ei5ndtse5hobry459pyasfhmf3clhur 2 250057 2835623 2835112 2026-10-03T05:40:30Z Platos Cave (physics) 2562653 2835623 wikitext text/x-wiki {{Original research}} '''Simulating gravitational and atomic orbits via n-body rotating particle-particle orbital pairs at the Planck scale''' The following describes a geometrical method for simulating both gravitational orbits <ref>Macleod, Malcolm J.; {{Cite journal |title=3. Gravitational orbits from n-body rotating particle-particle orbital pairs |journal=SSRN |date=Dec 2021 | doi=10.2139/ssrn.3444571}}</ref><ref>https://codingthecosmos.com/podcasts_mathematical_electron_model.html Podcast 3. Gravitational orbitals</ref> and atomic orbitals <ref>Macleod, Malcolm J.; {{Cite journal |title=4. Geometrical origins of quantization in H atom electron transitions |journal=SSRN |date=Dec 2022 | doi=10.2139/ssrn.3703266}}</ref><ref>https://codingthecosmos.com/podcasts_mathematical_electron_model.html Podcast 4. Atomic orbital transitions</ref> via an n-body network of rotating individual particle-particle orbital pairs (the atomic orbital transition is mapped as a single orbital pair). Although the simulation is dimensionless (the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]]), it can translate via the [[w:Planck_units |Planck units]] for comparisons with real world orbits. The orbits generated by this dimensionless geometrical approach can be formulated, and despite not using Newtonian physics these formulas demonstrate consistency; for example the derived formulas for radius R, period T and (M + m) will reduce Kepler's 3rd law to '''G'''. Likewise the atomic orbital shells naturally quantize according to pi without relying on built-in postulates. [[File:Gravitational-regular-3body-orbit.gif|thumb|right|640px|A regular 3-body orbit. The simulation begins with the start (x, y) co-ordinates of each point. No other parameters are required. r0=2*α; x1=1789.5722983; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0]] :<math>\frac{4 \pi^2 R^3}{(M+m) T^2} = \frac{l_p c^2}{m_P} = G</math> The simulation source code(s) used here are listed below, these give a precise description of this orbital model and so can be used as reference where explanation is inadequate <ref>https://simulationuniverse.org/Gravity_atomic_orbital_source-codes.html simulation source codes in C</ref>. For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''_P (for example, a 1kg satellite would be divided into 1kg/''m''_P = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). For simulating electron transition within the atom, the same program is used. The electron is assigned as a single point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' (a unit of length) is added to (increases) the electron - proton orbital radius in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although we are mapping the electron transition as a gravitational orbit on a 2-D plane, periodically the transition spiral angles converge to give an integer orbital radius (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''), a radial quantization (as a function of pi and so of geometrical origin) naturally emerges. Furthermore, the transition steps between these radius can then be used to solve the transition frequency, replicating the Bohr model. The gravitational orbit returns results analogous to the Bohr model and the 2-photon orbital model approach <ref>Macleod, Malcolm J.; {{Cite journal |title=4. Geometrical origins of quantization in H atom electron transitions |journal=SSRN |date=Dec 2022 | doi=10.2139/ssrn.3703266}}</ref> complements the (electric wave-state) [[w:Schrodinger equation |Schrodinger wave equation]]. [[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point accordingly, we can 'design' the required orbits. The 26 points orbit each other resulting in 325 point-point orbitals.]] === Theory === In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is a continuous oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10²³ units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the mathematical electron model<ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>. If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10²³ units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10²³ electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity is a mass to mass interaction. Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map individual particle to particle orbitals albeit as gravitational orbits, the H atom is a well-researched particle-to-particle orbital pair (an electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the 2-photon-orbital approach, in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). Orbital momentum derives from this orbital radius, it is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a hyperbolic spiral for which the angle component periodically converges to give integer radius where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this particular quantization has a geometrical origin. Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the general formula <math>f/c = t \lambda_H/(n_f^2-n_i^2)</math>. In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The transition spiral provides the geometric structure; the photon-orbital hybrid (2-Photon model) provides the angular momentum content. === N-body orbitals === [[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]] The simulation universe is a 4-axis hypersphere expanding in increments <ref>Macleod, Malcolm; {{Cite journal |title=1. Planck unit scaffolding correlates with the Cosmic Microwave Background |journal=SSRN |date=26 March 2014 | doi=10.2139/ssrn.3333513 }}</ref> with 3-axis (the [[v:User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'. The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m_P]], [[w:Planck length |Planck length l_p]], [[w:Planck time |Planck time t_p]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''_p/''t''_p) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. Setting <math>\alpha_{inv} = \frac{1}{\alpha}</math> = 137.035999177⁠ (CODATA 2022) Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). :<math>\beta_{gravity} = \frac{1}{r_{ij} r_{orbital} \sqrt{r_{orbital}}}</math> :<math>\beta_{atomic} = \frac{1}{\sqrt{2\alpha_{inv}} r_{orbital} \sqrt{r_{orbital}}}</math> As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation. Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] <ref>Macleod, Malcolm; {{Cite journal |title=2. Relativity as the mathematics of perspective in a hyper-sphere universe |journal=SSRN |date=26 March 2014 | doi=10.2139/ssrn.3334282 }}</ref> is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence. [[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]] ==== 2 body orbits ('''x, y''' plane) ==== For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units :<math>r_s = \frac{2 l_p M}{m_P}</math> As the simulation itself is dimensionless, we can remove the dimensioned length component <math>2 l_p</math>, and as each point is analogous to 1 unit of Planck mass <math>m_P</math>, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius <math>r_s</math> = number of central points = total points - 1 (the orbiting point). After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. '''Key''': 1. '''i''' = r<sub>s</sub>; the number of center mass points (the orbited object). 2. '''j''' = total number of points, as here there is only 1 orbiting point; '''j''' = '''i''' + 1 3. '''k<sub>r</sub>''' is a mass to radius co-efficient in the form <math>j_{max} = (k_r i + 1)</math>. This function defines orbital radius in terms of the central mass Schwarzschild radius (<math>i</math>) and the orbiting point (1), thus quantizing the radius. When <math>k_r</math> = 1 then <math>j_{max} = j</math>, and the radius is at a minimum giving an analogue gravitational principal quantum number <math>n_g = j_{max}/j</math>. 4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 5. '''r<sub>α</sub>''' = a radius constant, here r<sub>α</sub> = sqrt(2/α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals. :<math>r_{\alpha} = \sqrt{2\alpha_{inv}} </math> :<math>r_{orbit} = {r_{\alpha_{inv}}}^2 \;*\; r_{wavelength} </math> 6. Rotation angle β :<math>\beta_{orbital} = \frac{1}{r_{ij} r_{orbital} \sqrt{r_{orbital}}}</math> :<math>r_{ij} = \sqrt{\frac{2 j}{i}}</math> (for each gravitational orbital in the simulation) :<math>r_{ij} = \sqrt{2 \alpha_{inv}}</math> (for each atomic orbital in the simulation) ==== Orbital formulas (2-D plane)==== :<math>j = i + 1</math> :<math>r_{orbit} = 2 \alpha_{inv} 2 \frac{(k_r i + 1)^2}{i^2}</math>, orbital radius (center mass to point) :<math>r_{ij} = \sqrt{\frac{i}{j}}</math> (averaged for each orbit) :<math>t_{orbit} = \frac{2\pi}{ \beta_{orbit}} = 16 \pi {\alpha_{inv}}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} j^{1/2}}</math>, orbiting point period :<math>r_{barycenter} = \frac{r_{orbit}}{j}</math> :<math>l_{orbit} = 2 \pi (r_{orbit} - r_{barycenter})</math>, distance travelled by orbiting point :<math>v_{orbit} = \sqrt{\frac{i}{r_{orbit}j}}</math>, orbiting point velocity Example (dimensionless). The simulation parameters agree closely with the calculated parameters: :points = 8 (1 orbiting point and 7 center mass points) <ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Gravitational-orbital-simulation-2body.c</ref> :i = 7, j = 8 :k<sub>r</sub> = 32 :<math>\sqrt{2j/i}</math> = 1.511858 Calculated :calculated orbit period = 2504836149.00059 :calculated orbit radius = 566322.241497 :calculated orbit length = 3113519.13854 :calculated orbit barycenter = 70790.280187, 0 :n<sub>g</sub> = (k<sub>r</sub> i + 1)/j = 28.125 Simulation simulation orbit period = 2504836141 simulation orbit length = 3113519.130546298 simulation orbit barycenter; x = 70790.28092, y = 0.000732 ===== Earth moon orbit===== The earth to moon mass ratio approximates 81:1 and so can be simulated as a 2-body orbit with the moon as a single orbiting point as in the above example. Here we use the orbital parameters to determine the value for the mass to radius coefficient k<sub>r</sub>. Planck length <math>l_p</math>, Planck mass <math>m_P</math> and <math>c</math> are used to convert between the dimensionless formulas and dimensioned SI units. Reference values :<math>M</math> = 5.9722 x 10<sup>24</sup>kg (earth) :<math>m</math> = 7.346 x 10<sup>22</sup>kg (moon) :<math>T_{orbit}</math> = 27.321661*86400 = 2360591.51s To simplify, we assume a circular orbit which then gives us this radius :<math>R_{orbit} = (\frac{G (M+m) T_{orbit}^2}{4 \pi^2})^{(1/3)}</math> = 384714027m :<math>G = \frac{l_p c^2}{m_P}</math> = 0.66725e-10 The mass ratio :<math>i = \frac{M}{m}</math> = 81.298666, j = i + 1 We then find a value for <math>k_r</math> using T<sub>orbit</sub> as our reference (reversing the orbit period equation). :<math>T_o = T_{orbit} \frac{m_P}{M} \frac{c}{l_p} = 16 \pi {\alpha_{inv}}^{3/2} \frac{(k_r i + 1)^3}{i^{5/2} j^{1/2}}</math> (dimensionless orbital period) :<math>k_r = \frac{1}{i} {(\frac{T_o i^{5/2} j^{1/2}}{16 \pi {\alpha_{inv}}^{3/2}})}^{(1/3)} - \frac{1}{i}</math> = 12581.4468 Dimensionless solutions :<math>r_{orbit} = 2 \alpha_{inv} 2 \frac{(k_r i + 1)^2}{i^2}</math> = 86767420100 :<math>t_{orbit} = 16 \pi {\alpha_{inv}}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} j^{1/2}}</math> = 0.159610040233 x 10<sup>18</sup> :<math>r_{barycenter} = \frac{r_{orbit}}{j}</math> = 1054299229.62 :<math>l_{orbit} = 2 \pi (r_{orbit} - r_{barycenter})</math> = 538551421685 :<math>v = \sqrt{\frac{i}{r_{orbit}j}}</math> = 0.33741701 x 10<sup>-5</sup> Converting back to dimensioned values :<math>R = r_{orbit} l_p \frac{M}{m_P} = R_{orbit}</math> = 384714027m :<math>T = t_{orbit} \frac{l_p}{c} \frac{M}{m_P} = T_{orbit}</math> = 2360591.51s :<math>B = \frac{R}{j}</math> = 4674608.301m (barycenter) :<math>L = 2\pi (R - B)</math> = 2387858091.51m (distance moon travelled around the barycenter) :<math>V = c \sqrt{\frac{i}{r_{orbit}j}}</math> = 1011.551m/s (velocity of the moon around the barycenter) If we expand the velocity term :<math>v_{orbit} = c \sqrt{\frac{i}{r_{orbit}j}}</math> :<math>v_{orbit}^3 = \frac{G M}{T_{orbit}} 2\pi \frac{i^2}{j^2}</math> Note: The [[w:standard gravitational parameter | standard gravitational parameter]] ''μ'' is the product of the gravitational constant ''G'' and the mass ''M'' of that body. For several objects in the Solar System, the value of ''μ'' is known to greater accuracy than either ''G'' or ''M''. :<math>\mu_{earth}</math> = 3.986004418(8)e14 :<math>\mu_{moon}</math> = 4.9048695(9)e12 :<math>i = \frac{\mu_{earth}}{\mu_{moon}}</math> = 81.2662685 :<math>k_r = \frac{c}{2 \sqrt{\alpha_{inv}}} {(\frac{T_{orbit}}{2 \pi \mu_{earth}})}^{1/3} \frac{(i+1)}{i}^{1/6} - \frac{1}{i}</math> = 12580.3462 :<math>t_{orbit}</math> = 0.15956776936 x 10<sup>18</sup> :<math>r_{orbit}</math> = 86752239934 ===== Kepler's formula = G===== Kepler's formula reduces to G :<math>R = 2 \alpha_{inv} 2 (\frac{k_r i + 1}{i})^2 l_p \frac{M}{m_P}</math> :<math>T = 16 \pi {\alpha_{inv}}^{3/2} \frac{{(k_r i + 1)}^3}{i^{5/2} (i+1)^{1/2}} \frac{l_p}{c} \frac{M}{m_P}</math> :<math>M+m = M (\frac{i+1}{i})</math> :<math>\frac{4 \pi^2 R^3}{(M+m) T^2} = \frac{l_p c^2}{m_P} = G</math> Maple code : R:=(2/alpha_{inv})*2*((kr*i+1)^2/i^2)*lp*(M/mP): : T:=(16*Pi/alpha_{inv}^(3/2))*((kr*i+1)^3/(i^(5/2)*(i+1)^(1/2)))*(lp/c)*(M/mP): : Mm:=M*(i+1)/i: : simplify(4*Pi^2*R^3/(Mm*T^2)); : Output: lp*c^2/mP [[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]] ==== Orbital alignment ==== Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit. The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). An orbital drift (as determined where the blue and yellow meet) naturally occurs; the eccentricity (shape) of the orbit a function of center mass and the ratio of alignment of the orbitals. A near straight line orbit will have a greater drift and a greater eccentricity than a near circular orbit. The elliptical orbit has a longer period than the circular orbit (which has a 360 degree orbit, the sidereal period). The additional period is known as the anomalistic period and includes the precession angle (360 + precession angle). Note: in these simulations there are only 2 orbital types; clock-wise and anti-clockwise ... in a real world orbit there will be a mixture. ===== Precession ===== Precession is a change in the orientation of the rotational axis of a rotating body. The first of three tests to establish observational evidence for the theory of general relativity, as proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the[[w:Tests_of_general_relativity | perihelion of Mercury]]. This precession is not predicted by Newtonian gravity. The formula for precession uses the semi-major axis ''a'' (the maximum distance between center of mass) and the semi-minor axis ''b'' (the minimum distance between center of mass). :<math>e = \sqrt{1-\frac{b^2}{a^2}}</math> :<math>\theta = \frac{6\pi G M}{a (1-e^2) c^2}</math> Where ''e'' is the eccentricity of the orbit and θ is the precession angle The frequency of the center mass Schwarzschild radius = <math>i 2 l_p</math>, where ''i'' is the number of Planck mass points in the center mass and ''l''<sub>p</sub> is Planck length; ''a'' and ''b'' become :<math>a = r_a i 2 l_p</math> :<math>b = r_b i 2 l_p</math> The formula for <math>r_a</math> :<math>r_a = 2 \alpha 2 \frac{(k_r i + 1)^2}{i^2}</math> For example, the precession angle for Mercury :<math>\theta = \frac{6\pi G M}{a (1-e^2) c^2} = \frac{6\pi \lambda_{sun} a}{2 b^2}</math> = 0.501866 x 10-6 radians The Schwarzschild radius of the sun <math>\lambda_{sun} = i 2 l_p</math> = 2953.25m The eccentricity of Mercury :<math>e = \sqrt{1-\frac{b^2}{a^2}}</math> = 0.203225 (where a = 57909050km and b = 56671523km) In this simulation the ratio of anti-clockwise:clockwise orbitals = 108:1 with orbiting mass = 1 mass unit (k<sub>r</sub> = 12) <ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Orbital model source code repository</ref>. {| class="wikitable" !Center mass!!angle θ!!θ*i!!eccentricity!!''r''<sub>a</sub>, ''r''<sub>b</sub> |- | 24 | 0.001175503 | 0.028212072 | 0.194749592 | 79481.8311615, 77959.9920879 |- | 28 | 0.001009240 | 0.028022688 | 0.195433743 | 79403.2724007, 77872.1317383 |- | 32 | 0.000884077 | 0.028290464 | 0.197440737 | 79344.3788203, 77782.4708225 |- | 36 | 0.000786489 | 0.028313604 | 0.197813449 | 79298.5878077, 77731.6227135 |- | 40 | 0.000708274 | 0.028330960 | 0.198373657 | 79261.9645140, 77686.7492830 |- | 44 | 0.000644252 | 0.028347088 | 0.199144931 | 79232.0062928, 77644.9930740 |- | 48 | 0.000590779 | 0.028357392 | 0.200476249 | 79207.0454340, 77599.0285567 |- | 52 | 0.000545493 | 0.028365636 | 0.200748008 | 79185.9277789, 77573.9329729 |- | ... | | | | |- | ∞ | 0.000005888 | | 0.205660603 | |} [[File:Ellipse-extrapolation.png|thumb|right|379px|Extrapolation of simulated elliptical data using the rotating orbital gravity simulator]] At a low mass ratio the mass influences the eccentricity, this influence reduces as mass increases and so the ratio 108:1 was chosen because extrapolating to infinity (the sun:mercury mass ratio = 6025000:1) gives an eccentricity ''e'' = 0.20566 close to that of the Mercury orbit ''e'' = 0.20563. Likewise the extrapolated precession angle = 0.000005888 is only slightly greater than the Mercury orbit angle θ = 0.000005019. Due to computational limitations, only a short radius ''r''<sub>a</sub> and low central masses (''M'' 24...52) were simulated. A longer radius and larger central mass values are required for greater precision. Extrapolating over this scale is imprecise however the it does appear that the correlation is within 1-2 orders of magnitude. For the simulation program code <ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Gravitational-orbital-elliptical-orbits.cpp</ref> and the extrapolation code used <ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Gravitational-orbital-ellipse-extrapolate.py</ref>. [[w:Frame-dragging |Frame dragging]] can also impact the results as the central mass is still of itself a gravitational orbit (the center points also orbit each other), and so the center mass rotates at a relatively high velocity when compared with the orbiting point. Lense-Thirring in dimensionless form; :<math>T = T_d \frac{m_P c}{M l_p}</math> :<math>R = R_d \frac{M l_p}{m_P}</math> :<math>J = J_d \frac{M R^2}{T}</math> :<math>\theta_f = \frac{2 G J T}{c^2 r^2} = 2\frac{J_d}{R_d}</math> ==== Gravitational coupling constant ==== In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α<sub>G</sub>'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio; :<math>\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}</math> For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α<sub>G</sub> then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>). :<math>{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}</math> As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α<sub>G</sub> is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10<sup>42</sup> units of Planck time (<math>t_p = 10^{-42}s</math>), this occurs about once every 3 minutes. :<math>\frac{{\alpha_G}^{-1}}{t_p}</math> Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak. If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f<sub>e</sub>, then the point will require 10<sup>23</sup> electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time). :<math>f_e = \frac{m_P}{m_e} = 10^{23}</math> For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have <math>1kg/m_P = 45940509</math> particles in the point-state. The earth (A) will have <math>5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}</math> particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite; :<math>N_{orbitals} = (\frac{m_A}{m_P})(\frac{m_B}{m_P}) = 0.1261\; x10^{41}</math> Earthe parameters: :<math>i = \frac{M_{earth}}{m_P} = 0.27444 \;x10^{33}</math> (earth as the center mass) :<math>i 2 l_p = 0.00887</math> (earth Schwarzschild radius) :<math>s = \frac{1kg}{m_P} = 45940509</math> (1kg orbiting satellite) :<math>N_{orbitals} = i*s = 0.1261 \;x10^{41}</math> The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). :<math>E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J</math> (energy per orbital) :<math>E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg</math> The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs. :<math>N_{orbitals} = \frac{M_{sun}}{m_P} \frac{M_{planet}}{m_P}</math> :<math>n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} </math> :<math>L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha_{inv}},\;\frac{kg m^2}{s} </math> The orbital angular momentum of the planets; mercury = .9153 x10<sup>39</sup> venus = .1844 x10<sup>41</sup> earth = .2662 x10<sup>41</sup> mars = .3530 x10<sup>40</sup> jupiter = .1929 x10<sup>44</sup> pluto = .365 x10<sup>39</sup> [[File:Newton-vs-Orbital-trajectory_comparison.png|thumb|right|479px|Comparison of Newtonian and Orbital model simulations]] ==== Orbital vs. Newton ==== A 3-body orbit <ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Newton-vs-Orbital_Newton.py</ref><ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html Newton-vs-Orbital_Orbital.py</ref> is compared with the equivalent orbit using Newtonian dynamics. The start positions are the same r0=2*α; x1=3490.3069; y1=0; x2=cos(pi*2/3)*r0; y2=sin(pi*2/3)*r0; x3=cos(pi*2/3)*r0; y3=sin(pi*2/3)*r0 The orbiting point was used to determine the optimal G for the Newtonian orbit (G = 0.4956). Period of orbit (<math>k_r</math> = 2.19006) :<math>t_{calc}</math> = 1122034 :<math>t_{orbital}</math> = 1121397 :<math>t_{newton}</math> = 1125633 [[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2<sup>32</sup> steps they have clumped to form 1 large mass and 2 orbiting masses.]] ==== Freely moving points ==== The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted. In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2<sup>32</sup> steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. [[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]] ==== Hyper-sphere orbit ==== {{main|User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity}} Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates <ref>Macleod, Malcolm; {{Cite journal |title=2. Relativity as the mathematics of perspective in a hyper-sphere universe |journal=SSRN |date=2014 | doi=10.2139/ssrn.3334282 }}</ref>. Time becomes [[v:User:Platos Cave (physics)/Simulation_Hypothesis/God_(programmer)#Universe_time-line |time-line]]. While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B<sup>1</sup> to B<sup>11</sup>) around the ''A'' time-line (vertical expansion) axis ('''t<sub>d</sub>''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. === Atomic orbitals === [[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]] ==== Gravitational orbit ==== In the context of the gravitational orbital pair model, the atomic orbital is treated as a 2-body orbit (the electron as a point orbiting an n-body central mass 'nucleus'). To simulate electron transition between orbitals, at each step a unit of length is added. As the electron still continues its orbit, instead of a circular path, it traces a spiral path outwards. Notes: 1. Seen from the wave-point oscillation cycle model, the electron is predominately in the electric wave-state (which is a locally undefined state), the simulation program can only calculate the mass point-states, and so we are mapping only the gravitational orbital component of the electron orbit. For the gravitational orbit, the orbit co-ordinates were updated per unit of Planck time (with orbitals rotating 1 unit of Planck length). In the atomic orbital we have only 1 point (the electron), and it occurs as a point only after every wavelength cycle, and so we are updating our orbit only after every wavelength cycle. For example we find that at the lowest (''n'' = 1) orbit, there are about 471964 wave-point oscillations, and so we will map a polygon with 471964 sides (the wave state cannot be mapped). To compensate, the orbital rotation angle is modified such that 471964 steps are required to map 1 complete orbit ; :<math>\beta_{orbital} = \frac{1}{r_{ij} r_{orbital} \sqrt{r_{orbital}}}</math> :<math>r_{ij} = \sqrt{2 \alpha}</math> 2. During transition, at each step 1 unit of the photon is absorbed by the orbital radius, this continues until the photon is completely absorbed, the orbital radius extending proportionately. Photon absorption is not instantaneous, but instead occurs over time according to the photon wavelength. The number of steps required for this absorption then gives us the transition frequency. Notably, at integer (''Bohr'') radius (4''r'', 9''r'', 16''r''...) the spiral angle is a function of pi (2pi, (8/3)pi, 3pi ...), and as this is a linear extrapolation (the orbital radius grows in steps), ''n'' level orbits can be interpreted as specific orbits (which are inherently stable) occurring along the semi-continuous transition orbital path traced by the electron. The electron does not jump between n-levels, but the interim state (the transition phase) between these quantized n-levels is an orbital-photon hybrid state and thus difficult to detect. We may then have the equivalent of a non-integer ''n'', but as it is interchangeable with spiral angle and radius, we need only 1 parameter to calculate the other 2, and as transition frequency can also be determined in terms of angle and radius, we have a simple geometrical model that provides a framework for the transition process yet uses only pi and alpha as inputs. ==== 2-Photon orbital model ==== The Lyman series energy formula can be decomposed: :<math>\frac{1}{\lambda} = R_\infty\left(1 - \frac{1}{n^2}\right) = R_\infty - \frac{R_\infty}{n^2}</math> Mathematically (if not physically) we can divide into 2 waves :<math>\text{Photon}_{n1} = R_\infty</math> :<math>\text{Photon}_{nfinal} = (- \frac{R_\infty}{n^2})</math> :<math>\text{Photon}_{\text{total}} = \text{Photon}_{n1} + (-\text{Photon}_{n\text{final}})</math> This (mathematical) approach permits us to divide the transition into two distinct geometric processes taking place between the incoming photon and the orbital radius, with the electron taking a relatively passive role. Rather than 2 actual distinct photons, we may presume two geometric phases of a single photon absorption, nevertheless the 2-photon image is easier to conceptualize. Note these processes are not instantaneous but rather occur over time in discrete steps; '''Process 1 (Cancellation):''' A photon with energy corresponding to the ''n''=1 orbital frequency '''cancels''' the existing orbital structure. :<math>\text{Photon}_{n1} + \text{Orbital}_{n1} == zero</math> '''Process 2 (Creation):''' A (-) photon with energy corresponding to the <math>n_{\text{final}}</math> orbital '''creates''' the new orbital structure. :<math>\text{Photon}_{nfinal} == \text{Orbital}_{nfinal}</math> During transition, the system exists in a photon-orbital hybrid state. This is not a quantum superposition but a geometric intermediate configuration where: : The incoming photon’s momentum is being transferred incrementally : The orbital structure is simultaneously being dismantled and reconstructed : The electron mediates the momentum transfer through its position : Each steps transfers a quantum of momentum The gravitational orbit returns results analogous to the Bohr model and the 2-photon approach complements the (electric wave-state) Schrodinger equation <ref>Macleod, Malcolm J.; {{Cite journal |title=4. Geometrical origins of quantization in H atom electron transitions |journal=SSRN |date=Dec 2022 | doi=10.2139/ssrn.3703266}}</ref>. ==== Theory ==== =====Hyperbolic spiral===== [[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]] A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). For the particular spiral that the electron transition path maps, periodically the spiral angles converge to give integer radius, the general form for this type of spiral (beginning at the outer limit ranging inwards); :<math>x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 < \varphi < 4\pi</math> :radius = <math>\sqrt(x^2 + y^2) r</math> :<math>\varphi = (2)\pi, \; 4r</math> (360°) :<math>\varphi = (4/3)\pi,\; 9r</math> (240°) :<math>\varphi = (1)\pi, \; 16r</math> (180°) :<math>\varphi = (4/5)\pi, \; 25r</math> (144°) :<math>\varphi = (2/3)\pi, \; 36r</math> (120°) [[File:Bohr atom model (mul).svg|thumb|right|320px|Electron at different ''n'' level orbitals]] =====Principal quantum number '''n'''===== The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level. The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''² (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''² electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized. =====(Bohr) orbital===== Setting <math>\alpha_{inv} = \frac{1}{\alpha}</math> = 137.035999177⁠ (CODATA 2022) The Bohr radius has 2 components, the dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and the dimensioned (electron wavelength); :<math>a_0 = \alpha_{inv} \times \lambda_e</math> The orbital radius of this model also includes the proton wavelength and so is slightly greater than 2x the Bohr radius :<math>\lambda_0 = \lambda_p + \lambda_e</math> :<math>r_0 = 2 \alpha_{inv} \times \lambda_0</math> To reduce simulation computing time, only the alpha component is described here, the dimensioned components added later. :<math>r_{orbital} = 2 \alpha_{inv} n^2</math> As a mass point, the electron orbits the proton at a fixed radius (<math>r_{orbital}</math>) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius. [[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]] length = <math>l_{step} = \frac{1}{r_{orbital}}</math> Duration = 1 step per wavelength and so velocity velocity = <math>v_{orbital} = v_{step} = \frac{1}{2\alpha_{inv} n}</math> Giving period of orbit period = <math>t_{orbital} = \frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha_{inv} 2\alpha_{inv} n^3 </math> The (reference) orbital (''n'' = 1) :<math>t_{n1} = 2\pi 4 \alpha_{inv}^2</math> = 471964.356... The angle of rotation depends on the orbital radius :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha_{inv}}}</math> ===== 2-Photon orbitals ===== The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the 2-photon orbital model <ref>Macleod, Malcolm J.; {{Cite journal |title=4. Geometrical origins of quantization in H atom electron transitions |journal=SSRN |date=Dec 2022 | doi=10.2139/ssrn.3703266}}</ref>, the orbital (Bohr) radius is treated as a 'physical wave' (has physical properties) akin to the photon albeit an inverse such that <math>orbital \;radius + photon = zero</math> (they cancel). As such it is the orbital radius itself that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron has a `passive' role in the transition phase (mediating the exchange of momentun between the orbital and the photon). It is the rotation of the orbital radius that drives the electron motion, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues also during the transition phase resulting in the electron following a hyperbolic spiral path. ====== Spiral (mass point) ====== The photon is divided into 2 photons as per the Rydberg formula (denoted initial and final). :<math>\lambda_{photon} = (+\lambda_i) - (+\lambda_f)</math> The wavelength of the (<math>\lambda_i</math>) photon corresponds to the wavelength of the orbital radius. The (+<math>\lambda_i</math>) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-<math>\lambda_f</math>), because of the Rydberg minus term, will be equivalent to the orbital radius, and so conversely will increase the orbital radius. And so for the duration of the (+<math>\lambda_i</math>) photon wavelength, the orbital radius does not change as the 2 photons cancel each other; :<math>r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)</math> However, the (<math>\lambda_f</math>) has the longer wavelength, and so after the (<math>\lambda_i</math>) photon has been absorbed, and for the remaining duration of this (<math>\lambda_f</math>) photon wavelength, the orbital radius will be extended until the (<math>\lambda_f</math>) is also absorbed (the '''transition phase'''). For example, the electron is at the ''n'' = 1 orbital. To jump from an initial <math>n_i = 1</math> orbital to a final <math>n_f = 2</math> orbital, first the (<math>\lambda_i</math>) photon is absorbed by the orbital radius (<math>\lambda_i + \lambda_{orbital} = zero</math>), but simultaneously the (-<math>\lambda_f</math>) photon adds to the orbital radius, and so the electron follows a normal ''n'' = 1 orbit (the '''orbital phase'''), then the remaining (<math>\lambda_f</math>) photon continues until it too is absorbed (the '''transition phase'''). :<math>\lambda_i = 1 \times t_{n1}</math> (initial orbital) :<math>\lambda_f = n^2 \times t_{n1}</math> (final orbital) After the (<math>\lambda_i</math>) photon is absorbed, the (<math>\lambda_f</math>) photon still has <math>\lambda_f = (n_f^2 - n_i^2)t_{n1} = 3 t_{n1}</math> steps remaining until it too is absorbed. [[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]] This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of <math>r_{incr}</math> (both photon and orbital radius may be considered as constructs from multiple units of this geometry); :<math>r_{incr} = -\frac{1}{2 \pi 2\alpha_{inv}}</math> At each step 1 unit of <math>r_{incr}</math> is transferred from each photon to the orbital radius. During the orbital phase the <math>r_{incr}</math> unit from the (<math>\lambda_i</math>) photon is canceled by the (-)<math>r_{incr}</math> unit from the (<math>\lambda_f</math>) photon (<math>r_{incr}</math> + (-)<math>r_{incr}</math> = zero) and so the orbital radius does not change. After the (<math>\lambda_i</math>) photon is fully absorbed, the orbital radius continues to absorb (-)<math>r_{incr}</math> units from (what remains of) the (<math>\lambda_f</math>) photon, and as the (<math>\lambda_f</math>) photon is equivalent to the orbital radius, the orbital radius extends in these steps as a consequence. The number of steps <math>N_{steps}</math> required determines (and so can be used to calculate) the transition frequency. Furthermore at each step the orbital radius itself continues to rotate, the electron, being pulled along by this rotation according to angle β, thereby traces a spiral path as the orbital radius length changes. :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} r_{\alpha}}</math> The (accumulated) spiral angle <math>\Phi</math> at each <math>n^2</math> :n = 1 to 2: <math>\Phi = 2\pi \; (r = 4 \times \; r_{orbital}) </math> :n = 1 to 3: <math>\Phi = \frac{8}{3} \pi \; (r = 9 \times \; r_{orbital}) </math> :n = 1 to 4: <math>\Phi = 3\pi \; (r = 16 \times \; r_{orbital}) </math> :n = 1 to 5: <math>\Phi = \frac{16}{5}\pi \; (r = 25 \times \; r_{orbital}) </math> :n = 1 to 6: <math>\Phi = \frac{10}{3}\pi \; (r = 36 \times \; r_{orbital}) </math> :n = 1 to 7: <math>\Phi = \frac{7}{4}\pi \; (r = 49 \times \; r_{orbital}) </math> :n = 1 to 8: <math>\Phi = \frac{7}{2}\pi \; (r = 64 \times \; r_{orbital}) </math> In the mass state we can divide the photon into 2 mathematical structures, the transition frequency is defined as the inverse of one oscillation period at the Compton scale, multiplied by the geometric phase factor (including the dimensioned terms). This gives us the solution for the transition frequency and the Bohr radius with a pi-based spiral acting as the transition 'guard-rail'. We are using only alpha and pi, yet the transition frequency can be solved with high precision using the gravity simulator. This gives a geometrical derivation for the Bohr model, which in this context is the gravitational component of the electron orbit. ====== Photon-Orbital Hybrid (wave-state) ====== In this model, atomic orbitals are '''physical rotating structures''' rather than probability distributions. The orbital possesses: :Angular momentum: Quantized by discrete rotation steps :Energy: Stored in the rotating configuration The orbital rotates in discrete steps (β radians per step) rather than continuously. During transition, the system exists in a '''photon-orbital hybrid state'''. This is not a quantum superposition but a '''geometric intermediate configuration''' where: :The incoming photon's momentum is being transferred incrementally :The orbital structure is simultaneously being dismantled and reconstructed :The electron mediates the momentum transfer through its position :Each step transfers a quantum of momentum We then note. '''Standard QM''': Anti-realist---the wavefunction is a calculation tool, not a physical entity. Reality emerges only upon measurement. '''2-photon model''': Realist---orbitals are real rotating structures. The electron follows definite trajectories, even when unobserved. The geometric model thus establishes a '''two-layer architecture''' for encoding quantum states: :Geometric Framework Layer: The hyperbolic spiral structure provides the spatial scaffold <math>(r, \theta, \phi)</math> coordinates with built-in radial quantization <math>r \propto n^2</math> :Photon Information Layer: The absorbed photon carries angular momentum quantum numbers <math>(l, m_l)</math> that modulate this scaffold This separation offers that the Bohr model is not replaced by the Schrodinger equation but rather complements it (the mass-state provides the geometric structure; the photon-orbital hybrid provides the angular momentum content). ===== Simulation ===== [[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]] The gravity simulation (<ref>https://codingthecosmos.com/Gravity_atomic_orbital_source-codes.html source code</ref>) was setup with the electron as the orbiting point and 65 mass points to represent the nucleus (although the proton-electron mass ratio is 1836:1, this would be become a 1837-body set of orbiting points which is too computationally intensive). The program was modified with the rotational angle β as :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} r_{\alpha}}</math> To simulate the transition phase, at each step 1 unit of <math>r_{incr} = \frac{1}{2 \pi 2\alpha_{inv}}</math> was added to the proton-electron orbital radius. To convert to frequency in Hz, the dimension components were added :<math>f_H = \frac{2c}{\lambda_e + \lambda_p}</math> = 0.155184298 10²²s^-1 For n = 1 to n = n² :<math>f_n = 2\pi f_H \;\frac{n^2 - 1}{N_{steps}} </math> '''Results''' {| class="wikitable" !<math>r/r_{orbital}</math> !! <math>r/r_{orbital}</math> !! N-steps !! <math>\Phi</math> (deg) !! frequency Hz |- | 4.000000115 | 2.000004018 | 1887860.649 | 0.000017120 | 2466034304131826.5 |- | 8.999994875 | 4.000003286 | 4247681.247 | 120.000001964 | 2922708926063928.0 |- | 15.999987119 | 6.000002004 | 7551428.532 | 180.000002514 | 3082545855782738.5 |} Experimental values for H(1s-ns) transitions. :H(1s-2s) = 2466 061 413 187.035 kHz <ref>http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf</ref> :H(1s-3s) = 2922 743 278 665.79 kHz <ref>https://pubmed.ncbi.nlm.nih.gov/33243883/</ref> :H(1s-4s) = 3082 581 563 822.63 kHz <ref>https://codata.org/</ref> Although the simulation was a standard gravitational orbit (charge-less) and without a relativistic correction, the results were quite close; :n = 1 to n = 2: margin = 100 * (experimental - simulated)/experimental = 0.001099% :n = 1 to n = 3: margin = 0.001175% :n = 1 to n = 4: margin = 0.001158% ====== Rydberg atom ====== Here a Rydberg atom is defined as an (idealized) atom with a nucleus of point size and an infinite mass and so the barycenter = atom center at co-ordinates (0, 0). We can simulate this as a simple transition beginning at the initial (''n''_i = 1) orbital. :<math>\varphi = 0</math> (start angle) :<math>r_{orbital} = r_0 = 2\alpha</math> (nb. Bohr radius = <math>\alpha \lambda_e</math> = α*electron wavelength) :<math>x = r_{orbital},\; y = 0</math> (start co-ordinates) For each step during transition, setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and <math>n_f</math>. :<math>r = r_{orbital} + \frac{t}{2\pi 2\alpha}</math> (number of increments ''t'' of <math>r_{incr}</math>) :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}</math> :<math>\varphi = \varphi + \beta</math> :<math>n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}</math> (<math>n_f^2</math> as a function of ''t'') The spiral angle and <math>n_f^2</math> are interchangeable :<math>\varphi = 4 \pi (1 - \frac{1}{{n_f}^2})</math> We can then re-write (<math>n_f</math> is only an integer at prescribed spiral angles); :<math>\beta = \frac{1}{{r_{orbital}}^2 n_f^3}</math> ''n'' orbital transition periods are multiples of the ''n'' = 1 orbital. If we include a relativistic term; :<math>t_{n1} = 2\pi 4 \alpha_{inv}^2 \sqrt{(1 - \frac{1}{4 \alpha_{inv}^2})}</math> = 471961.21478 Then we can compare a theoretical Rydberg (also without including charge) and find a further improvement. For ionization, the second photon is of such long wavelength that its momentum contribution is negligible, and so the ionization transition frequency should correspond closely to the ''n'' = 1 orbital. :H(1s-∞s) = 3288 086 857 127.60 kHz <ref>https://codata.org/ (109678.77174307cm-1)</ref> (''n'' = ∞) This would give us a H(n1) = 471959.2427762 :n = 1: margin = 100 * (H(n1) - t_{n1})/H(n1) = 0.001099% :n = 1 to n = 2: margin = 100 * (experimental - 4 t_{n1})/experimental = 0.000267% :n = 1 to n = 3: margin = 100 * (experimental - 9 t_{n1})/experimental = 0.000398% :n = 1 to n = 4: margin = 100 * (experimental - 16 t_{n1})/experimental = 0.000422% If we use the non-relativistic <math>t_{n1}</math>, then our results are remarkably close to the results from the gravity simulation suggesting that central mass size does not have the major role in determining the transition frequency. :n = 1 to n = 2: margin = 0.000932% :n = 1 to n = 3: margin = 0.001063% :n = 1 to n = 4: margin = 0.001088% ====== Barycenter correction ====== The experimental hydrogen transition frequencies deviate slightly from the ideal Rydberg formula. When normalized, these deviations exhibit a characteristic pattern: a minimum near ''n'' = 2, followed by a rising trend toward the ionization limit as the radius increases. :H(n1) = 471959.242776 :H(n2) = 471959.956565 :H(n3) = 471959.338749 :H(n4) = 471959.222066 The gravity simulation results showed a similar trend suggesting in both cases a barycenter 'wobble'. :sim(n1) = 471964.35638 :sim(n2) = 471965.14481 :sim(n3) = 471964.88601 :sim(n4) = 471964.68921 Therefore, given the possibilities of this gravity simulator as a research tool, the next step is to divide the proton into 3 parts resembling the 3 quarks, and then experiment with different configurations. For example, we could form a U quark cluster and a D quark cluster and then test different arrangements of the clusters (and also different mass ratios) to determine which, if any, reduces the margin. This could give potential insights into the protons internal structure. We can also give the electron further mass points to determine if it also becomes constrained and relaxed. ==Article series== The following are HTML transcripts of journal articles * [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page ===The 3 levels=== A model overview from the sub Planck scale to the container universe * [[https://codingthecosmos.org/1a-CMB_from_Planck_unit_scaffolding.html 1a. Planck unit CMB.html]]: Part 1. A CMB from Planck units * [[https://codingthecosmos.org/1b-Sub_Planck_scale_from_tau_loop.html 1b. The Minimal Complexity Algorithm of the Planck Scaffolding]]: Part 2. The sub-Planck scale and the tau-loop * [[https://codingthecosmos.org/1c-the_Heavens_a_container_universe.html 1c. the Heavens.html]]: Part 3. The Heavens (the container universe) ===General articles=== * [[https://codingthecosmos.org/2-Relativity_as_mathematics_of_perspective.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective * [[https://codingthecosmos.org/3-Gravity_via_n-body_particle_orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs * [[https://codingthecosmos.org/4-Atomic-orbital_transition_using_Gravity_simulator.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs * [[https://codingthecosmos.org/5-Atomic_transition_via_photon_orbital_hybrid.html 5-w_axis.html]]: Imaginary number axis (radiation domain) * [[https://codingthecosmos.org/6-Natural_Planck_units_MTP_Fine_Structure_Constant_alpha.html 6-Physical-constant-anomalies.html]]: Derivation of the fine structure constant alpha from physical constant anomalies * [[https://codingthecosmos.org/7-Mathematical_electron_magnetic_monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as magnetic monopoles * [[https://codingthecosmos.org/8-Holographic_universe_2D_plus_Compton_wavelength.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue ===Wiki series=== Introductions to principle concepts (for more detailed discussion see General articles) * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron_(mathematical)]]: Mathematical electron formula * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass * [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model == External links == * [https://codingthecosmos.com/ Planck scale modelling using geometrical objects] * [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark (Book) * [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article) * [https://plato.stanford.edu/entries/pythagoras/ Pythagoras "all is number"] - Stanford University * [[w:Mathematical universe hypothesis | Mathematical universe hypothesis]] * [[w:Philosophy of mathematics | Philosophy of mathematics]] * [[w:Philosophy of physics | Philosophy of physics]] * [[w:Platonism | Platonism]] ==References== {{Reflist}} __INDEX__ own57wtc81z5cw511fg3t1mjg1dg9uq 0 263444 2835584 2756285 2026-10-02T22:59:08Z Scogdill 1331941 /* Timeline */ 2835584 wikitext text/x-wiki == Overview == Louisa (or Luise) Friederike Auguste Gräfin von Alten first married William Drogo Montagu, 7th Duke of Manchester when she was 20 and he was 28. He was one of the only '''33 dukes''' in the UK at this time, but not one of the wealthiest ones. Louise and Spencer Compton, Marquis of Hartington knew each other socially for a long time and began a sexual relationship, probably, in early 1863 or late 1862, after he returned from the U.S. and after her last child was born.<ref name=":1" />{{rp|p. 26}} The Duke of Manchester died in 1890, freeing her to pursue a more formal and open relationship with Hartington, who was created Duke of Devonshire the next year, in 1891. They married in August 1892. Louise was the Duchess of Devonshire between 1892 and 1911. The Duke of Devonshire died in 1908, the title going to his nephew, and she was Dowager Duchess until her death in 1911. Louise and Devonshire hosted their famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London on 2 July 1897. Louise was the subject of much comment and gossip in her life, arising in large part from her prominence in society and to a lesser degree in part from her conduct, which was not particularly concerned with middle-class notions of respectability. Referring to her long-standing relationship with Hartington, later the Duke of Devonshire, Henry Vane says Louise "was said to be the most powerful person outside the government":<blockquote>Louise Alten, for her part, exemplifies another vanished feature [of Victorian England], the role of the political hostess. The part she played in Hartington's career even while she was his mistress gives a special twist to this function, and her more conventional ascendancy as Duchess of Devonshire, when she was said to be the most powerful person outside the government, shows the influence that could be wielded by women in a political system from which they were formally excluded.<ref name=":20">Vane, Henry. ''Affair of State: A Biography of the 8th Duke and Duchess of Devonshire''. Peter Owen, 2004.</ref>{{rp|12}}</blockquote>Although she was fashionable and known for her beauty, she was individual and practical in her choices. She was wearing "scarlet tartan knickerbockers" under her hoops when she fell while climbing over a stile at a "paper-chase at Kimbolton in 1859."<ref name=":20" />{{rp|25}} (A paper chase was a "form of cross-country running in the early 19th century," also called "hare and hounds — the 'hares' started a few minutes before the others and left a trail of paper scraps to be followed by the 'hounds.'"<ref name=":30">{{Cite web|url=https://www.britannica.com/sports/cross-country|title=Cross-country {{!}} Training, Races & Benefits {{!}} Britannica|website=www.britannica.com|language=en|access-date=2025-08-11}}</ref>) According to Lady Eleanor Stanley, Louise's knickers, like those men wore when hunting, "were revealed to all the world in general and the Duc de Malakoff in particular" when she fell.<ref name=":20" />{{rp|25}} At a time when some women still wore undergarments that did not have a crotch, Louise was not exposed when she fell, however notable her knickerbockers may have been. (See [[Social Victorians/People/Louisa Montagu Cavendish#Timeline|1859 in the Timeline]] for the full quotation.) In the 1890s, probably (Louise was 60 in 1892), the French writer Elisabeth de Gramont says of her,<blockquote> I was presented to an English lady of importance, the Duchess of Devonshire. After the exchange of a few phrases, I was filled with consternation and said to myself: "I shall really have to alter my pronunciation, for the English I talk isn't at all the same as the Duchess of Devonshire's." I learned subsequently that she was a German, which reassured me. Omnipotent and very intelligent, she was the most representative member of the British aristocracy. Her name and her wealth, the political position of her two hus- [258–259] bands, the first and the one of the moment, made her something of an English Buddha. She was more than sixty years old when I saw her, but her age vanished beneath the aids of art: a wig made up of little fair curls framed the petal-pinkest of faces, a formidable iron corset kept her rigid and, although she took an hour to climb the stairs, she led in all respects the life of a very young woman. Balls, dinners, races, the Opera, social gatherings, charity-committees — she presided over and managed everything. She invited me to Devonshire House, a memorable spot renowned for its gorgeous entertainments, where for a long time the pulse of English high life beat strongest, a kind of pinnacle surmounted by tiaras and diadems. | was delighted at visiting that London palace, whose railings run for a good distance parallel to those of the Park in Piccadilly. I traversed suites of innumerable reception-rooms before running her to earth in a little room of which she seemed proud, her little French boudoir — half drawing-room, half dressing-room. She received me sitting in front of a Louis XV dressing-table. She was as unlike as [259–260] possible to that sketch of Gainsborough’s, which represents a powdered Duchess of Devonshire in boy’s disguise. The picture hangs at South Kensington. I never saw the Duchess of Devonshire again. For a long time she carried on the same representative existence, falling asleep in the end over her cards at bridge. When her husband died she suffered the fate of English dowagers. An iron curtain falls like a guillotine upon their life; in the twinkling of an eye they are brought low and stripped, and the new heirs get rid of them. They often depart for the Riviera. The Duchess of Devonshire retired to hotel with inclusive terms of three francs a day at Cimiez, near Nice, where she died.<ref name=":31">de Gramont, Elisabeth (Ex Duchesse de Clermont-Tonnere). ''Pomp and Circumstance''. Jonathan Cape & Harrison Smith, 1929. Internet Archive: https://archive.org/details/pompcircumstance0000degr/.</ref>{{rp|258–260}}</blockquote>In the spring of 1901, Consuelo, Duchess of Marlborough saw Louise, Duchess of Devonshire and describes her far more negatively:<blockquote>Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref name=":23" />{{rp|p. 115}}</blockquote> == Also Known As == *Louise, Duchess of Devonshire (15 January 1832 – 15 November 1911) *Louisa, Duchess of Manchester *Née Countess von Alten of Hanover *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *[[Social Victorians/People/Warwick|Daisy, Lady Warwick]] *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * [[Social Victorians/People/Marlborough|Consuelo Spencer-Churchill, Duchess of Marlborough]] (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} === '''Business Relationships''' === She had an account at Henry Poole & Co., 15 Saville Row, London, and used it to buy "a fancy silk vest (waistcoat) as a gift for the Prince of Wales."<ref>{{Cite web|url=https://henrypoole.com/individual/louisa-duchess-of-manchester/|title=The Double Duchess Louisa van Alten|website=Henry Poole Savile Row|language=en|access-date=2023-04-03}} https://henrypoole.com/individual/louisa-duchess-of-manchester/.</ref> == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu (later 7th Duke of Manchester) married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1858 March through 1859 June''', Louise, Duchess of Manchester was appointed Mistress of the Robes for Queen Victoria, during the Derby ministry.<ref>{{Cite web|url=https://www.unofficialroyalty.com/louise-montagu-duchess-of-manchester/|title=Louise Montagu, Duchess of Manchester|last=Scott|date=2018-11-01|website=Unofficial Royalty|language=en-US|access-date=2023-04-03}} https://www.unofficialroyalty.com/louise-montagu-duchess-of-manchester/.</ref> '''1859''' A paper chase is a cross-country race with the human "hares" leaving a paper trail for the "hounds"<ref name=":30" />:<blockquote>As for Louise, a glimpse of her unconventional ways is given by Lady Eleanor Stanley, describing a paper-chase at Kimbolton in 1859: "The Duchess of Manchester, in getting too hastily over a stile, caught a hoop of her cage in it, and went head over heels, alighting on her feet with her cage and whole petticoats remaining above her head. They say there was never such a thing seen — and the other ladies hardly knew whether to be thankful or not that a part of her underclothing consisted of a pair of scarlet tartan knickerbockers (the things Charles shoots in) — which were revealed to all the world in general and the Duc de Malakoff in particular." Lady Eleanor says that the other ladies hardly knew whether to be thankful because at this date a lady's normal wear under her petticoats was only a pair of leggings laced up to the waistband, an arrangement that could be unduly revealing in combination with a crinoline. ... But some of the more active ladies were beginning to adopt / Louise's expedient of donning a hidden pair of breeches for rambles in the country.<ref name=":20" />{{rp|25–26}}</blockquote>'''1862, late, or early 1863''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|William Douglas-Hamilton]] married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son, 8th Duke) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 January 28, Tuesday''', on this date, the ''Dundee Evening Telegraph'' reported that Louise, Duchess of Manchester [[Social Victorians/Timeline/1890#1890 January 28, Tuesday|had bought 30 "Exploration" shares]], which funded exploration expeditions with the possibility of rights to minerals or resources or the development of monopolies for goods that were "discovered."<ref>"Notes — Mainly Personal." ''Dundee Evening Telegraph'' 28 January 1890, Tuesday: 2 [of 4], Col. 2b [of my 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18900128&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''Evening Telegraph'', n.p.</ref> Her daughter in law, [[Social Victorians/People/Mandeville|Consuelo, Viscountess Mandeville]] also had bought 30 shares, both small amounts compared to some others. '''1890 March 22''', William Drogo Montagu (7th Duke and Louise's husband) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson, 9th Duke) and American heiress Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1891 December 21''', Spencer Compton Cavendish, Marquis of Hartington's father, the 7th Duke of Devonshire, died.<ref>"William Cavendish, 7th Duke of Devonshire." "Person Page — p958." ''The Peerage: A Genealogical Survey of the Aristocracy of Britain as Well as the Royal Families of Europe'' https://www.thepeerage.com/p958.htm (accessed 2022-10-13).</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1892, after''', French writer Elisabeth de Gramont met Louise, Duchess of Devonshire and describes her:<blockquote>a wig made up of little fair curls framed the petal-pinkest of faces, a formidable iron corset kept her rigid and, although she took an hour to climb the stairs, she led in all respects the life of a very young woman. Balls, dinners, races, the Opera, social gatherings, charity-committees — she presided over and managed everything.<ref name=":31" />{{rp|258–259}}</blockquote>'''1893, beginning in, on the Wednesday after Easter''', every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held.<p> '''1897 June 19, Saturday''', the Lady's Pictorial gossiping about the ball while claiming not to:. They're wrong about the costs of the dresses.</p><blockquote>The Duke and Duchess of Devonshire spent Whitsuntide at Chatsworth and went to an amateur theatrical performance at Buxton in aid of the Devonshire Hospital. The actors and actresses who were guests at Chatsworth were Lord Elcho, Mr. Leo Trevor, Mr. T. B. Mildmay, and Captain Jeffcock, Mrs. W. James, Mrs. Menzies, Miss Muriel Wilson, and Lady Mar and Kellie. The performance was a great success. Mr. A. J. Balfour was also a guest at Whitsuntide, and left later for Littlestone, where he enjoyed some golf. This week the Duke and Duchess were at Cowarth Park for Ascot. The Duchess of Devonshire's ball is a sensation second only to that of the great Commemoration procession itself. Her Grace herself is probably taking it more calmly than any one else. It is he ropenly expressed wish that nothing shall be written about it before it takes place, and as authoritative information regarding the special arrangements is being carefully withheld the publicity given to rumours on the subject is misjudged. The costumes are being prepared in London for by far the most part, not more than six are coming from Paris. The denouncements of the ball from the pulpit are neither original nor well founded, for it is gross exaggeration to state that any one dress will cost £2,000. Very, very few may cost one hundred, and the projected ball is doing much to help a season which, from a trade point of view, has up to now proved rather unsatisfactory. Doubtless it will be a great entertainment and a costly one, as well to givers as to guests. If, however, great people do not spend their money, what is to keep the ball rolling? If they save it, complaint is rife enough, and no fault can be found with the charitable celebration of this truly remarkable longest reign. That the Duchess of Devonshire is giving a really great costume ball as a social celebration of her Sovereign’s reign, being the longest in English hi tory [history], is a thing upon which her Grace is to be cordially felicitated. It is assured of success; many Royal personages will be present on the occasion, and are said to have intimated their intention of conforming to dress regulation.<ref>"Court & Society Notes." ''Lady's Pictorial'' 19 June 1897, Saturday: 65 [of 140], Col. 1c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005980/18970619/118/0065. Print title same, p. 911.</ref></blockquote> '''1897 July 2, Friday''', with the Duke of Devonshire, Louise Cavendish hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary Louise Elizabeth Montagu Douglas-Hamilton]] and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, [[Social Victorians/People/Marlborough|Consuelo Spencer-Churchill, Duchess of Marlborough]], describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps: <blockquote> A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref name=":23">Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}} </blockquote> The Duchess of Marlborough's description here, only 4 years after the ball, carries emotional content suggesting enmity between the two women. It may be merely uncharitably accurate, or it may be cruelly exaggerated in some way. '''1902 August 9''', just after King Edward VII's coronation Louise, Duchess of Devonshire tried "to reach the Ladies' before anyone else":<blockquote>After the long ceremony she tried to hurry out in the wake of the royal procession, but found herself stopped by a line of Grenadier Guards. Leonie [<nowiki/>[[Social Victorians/People/Leslie|Leonie Leslie]]] and Jennie [<nowiki/>[[Social Victorians/People/Churchill|Lady Randolph Churchill]]], who were descending from the King's special box, heard her upbraiding the officers in front of all the other peeresses, many of whom were themselves most uncomfortable. Then, trying to push her way past them, she missed her footing and fell headlong down a flight of steps to roll over on her back at the feet of the Chancellor of the Exchequer ([[Social Victorians/People/Hicks-Beach|Michael Hicks Beach]]), who stared paralyzed at this heap of velvet and ermine. The [[Social Victorians/People/de Soveral|Marquis de Soveral]] swiftly took charge of the situation and had her lifted to her feet while [[Social Victorians/People/Asquith|Margot Asquith]] nimbly retrieved the coronet, which was bouncing along the stalls, and placed it back on her head. It was a moment in which younger women naturally had to give precedence to an angry Duchess.<ref name=":28" />{{rp|190}}</blockquote> == The Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == Louise, Duchess of Devonshire and Spencer Compton, 8th Duke of Devonshire hosted the famous [[Social Victorians/Terminology#Fancy-dress ball|fancy-dress ball]], which took place on Friday night. According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>Such an event required extensive planning and attention to detail. The housekeeper aided by two secretaries was in charge of organisation inside the house with precise instructions from the Duchess on menus and all arrangements.<ref name=":18">Burlington, Julia, and Hamish Bowles. ''House Style: Five Centuries of Fashion at Chatsworth''. Skira Rizzoli, 2017. Based on an exhibition 25 March to 22 October 2017.</ref>{{rp|137}}</blockquote>At this time, we do not know who '''the housekeeper and two secretaries''' were. And where did this info about the "precise instructions" come from? === The Devonshire House Staff at the Ball === The Dukes of Devonshire did not keep employment records of the staff working at their various properties. The Servants and Staff Database managed by Chatsworth House has a list of everyone ever named in any documents as a servant or staff member,<ref name=":19" /> and by 1891, census records included employment at places like the properties owned by the Duke of Devonshire. == The Duchess of Devonshire's Costume == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} She was dressed as Zenobia, Queen of Palmyra. Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs The Album of Photographs|album presented to the Duchess of Devonshire]] and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted drop behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy included in the album is cropped so that those unfinished edges do not show, but it appears to be from the same photograph. That is, this photograph is the single source for our information about this portrait and the only contemporary image of her costume. The Lafayette Archive's copy of the actual negative of this photograph is of high-enough quality that a close-up of the Duchess's head shows more detail than can be seen otherwise. The page that includes it also has a photographic detail of a "star burst motif," possibly from an artifact at the British Museum, that might have influenced the making of the headdress: * Duchess of Devonshire, full-length, standing, 3/4 (Neg. No. 1350): http://lafayette.org.uk/dev1350.html * Another contemporary image, a line drawing from The Queen: More recent images of this costume also exist because it still exists in the collection in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) and has been exhibited and photographed in recent times. Even though the silver that is still on the dress has tarnished and turned dark and the jewels have been replaced, the costume is still striking. * Several new images of the costume are available as a result of the exhibit on the House of Worth in Paris, "Worth, Inventing Haute Couture": https://www.chatsworth.org/news-media/news-blogs-press-releases/queen-of-palmyra-ball-gown-on-display-in-paris/, which includes closeups of the fabric and embroidery. * An exhibit at Chatsworth in 2017 resulted in several images: ** blue mannequin, facing right, 3/4, wrong headdress: https://www.pinterest.com/pin/150589181285595564/ ** similar shot, from just a little farther back: https://www.pinterest.com/pin/443463894568152833/ ** blue mannequin, facing left, 3/4: https://www.pinterest.com/pin/443463894568152819/ ** no mannequin, facing left, 3/4, top half of dress, can see the inside of the shoulder: https://www.pinterest.com/pin/85146249199877896/ ** no mannequin, facing front, top half of dress: https://www.pinterest.com/pin/443463894568152807/ * The train ** An image of the green velvet train is here: https://royal-magazin.de/england/devonshire/zanobia-costume-headrdress-loise-duchess-of-devonshire.htm ** Another image of the train: https://www.pinterest.com/pin/37928821851398914/ ** Train from side and back: https://www.pinterest.com/pin/443463894568152809/ * A report about May Clyne's master's thesis on fancy-dress costumes in the 19th century says she was permitted to examine the dress. The article has one photograph plus a closeup (from the same photograph) of the costume on a mannequin.<ref>"Master's student explores rare iconic Victorian dress at Chatsworth Estate." Arts University Bournemouth. 29 November 2021. https://aub.ac.uk/latest/masters-student-explores-rare-iconic-victorian-dress-at-chatsworth-estate.</ref> [Perhaps we could interview Clyne or include her?] * "Debo," Deborah Cavendish, the 11th Duchess of Devonshire, wore Louise's costume for an event/ her birthday? ** With colored plumes, Simon Upton, photographer: https://interiorarchive.photoshelter.com/image?&_bqG=24&_bqH=eJzLrnIzDPKP9wuOTLQ09A0x9_EICQh0dHSMDzCwMjawMjI1sDI0AAIrz3iXYGfblNSy_LzijMyiVDWwQLyjn4ttCZAdGuwaFO_pYhsKUhxZ5h1YFB9Y6R5uoBbv6BxiW5yaWJScAQDEBSCo&GI_ID= ** "Debo" at the same event, journalist's photograph, closer and candid shot: https://www.pinterest.com/pin/532972937167697870/ === M. Attilio Comelli === [[File:Jones Zenobia.jpg|thumb|Costume design for Zenobia in ''The Masque of Queens'']] The Duchess's costume was designed by [[Social Victorians/People/Dressmakers and Costumiers#M. Comelli|M. Comelli]] and constructed by the [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|House of Worth]].<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> Because the collection in the archives of the Duke of Devonshire has Inigo Jones's designs for a Zenobia costume, which would have had 3 dominating colors,<ref>Simpson, Percy. "The Masque." In ''Shakespeare's England: An Account of the Life and Manners of His Age''. Vol. II. Oxford: Clarendon Press, 1916, p. 328. Internet Archive: [https://archive.org/details/shakespearesengl02raleuoft/page/n11/mode/2up?q=zenobia https://archive.org/details/shakespearesengl02raleuoft/].</ref> it is possible that Jones had an influence as well, if Comelli was familiar with them, although the Duchess of Devonshire's costume is not reminiscient of this design at all. Jones's 1608 design is for Zenobia in Ben Jonson's ''The Masque of Queens''.<ref>"File:Jones Zenobia.jpg." ''Wikimedia Commons'' (retrieved September 2025) https://commons.wikimedia.org/wiki/File:Jones_Zenobia.jpg.</ref> Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of the Duchess's retinue as well]]. Russell Harris says,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref> </blockquote> Comelli's renderings would be a fascinating addition to the information we have about the Duchess's costume. Such an addition might help us identify, for example, what Comelli designed, what the House of Worth added or changed and what accommodations to the design and constructed garment the Duchess required. The House of Worth handled not only the construction but also almost certainly any adjustments. The Duchess's body and opinions would have been central to perfecting the look and fit of this costume. The bodice and stomacher appear more loosely fitted than what many other women at the ball were wearing. As fashionable as it was, this costume would have been more comfortable than one that required tight-laced corsetry or than one whose drape and fit were more controlled. The Duchess likely had a great deal of say as well about the extremely ornate embellishment, since real jewels and pearls from her personal collection were taken out of their settings for this costume for the day of this ball and included in the spectacular trim. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focus on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or, less likely, a shared primary source: * The London ''Evening Standard'', ''Morning Post'' and ''Times'' published almost identical descriptions, varying only by the occasional punctuation mark or short phrase (e.g., "supplied by Worth, of Paris"). The ''Times'' was a morning paper, published at 5 a.m.<ref name=":27" />{{rp|56}}; the ''Morning Post'', despite its name, was published at 3:00 p.m.<ref name=":27" />{{rp|55}}; and the ''Evening Standard'' probably published later<ref name=":27" />{{rp|81}}. If one of these articles is the source for the others, it is likely the version published first, in the ''Times''. The fullest version is this one, from the ''Evening Standard'':<blockquote>The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond '''ornament'''. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a [[Social Victorians/Terminology#Stomacher|stomacher]] of real diamonds, rubies, and emeralds, and there was a jewelled belt.<ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}}</blockquote> *A morning paper, the ''Daily News'' has some of its highly detailed reportage wrong, including colors in the train and the number of people in the Duchess's entourage:<blockquote>The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar '''ornament'''. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. ... Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail.<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}}</blockquote> * The ''Daily Telegraph''<nowiki/>'s descriptions of the Duchess as Zenobia is the 2nd part of the opening of the story, imagining a narrative where she meets guests at the top of the staircase and then the Prince of Wales leads the Princess of Wales into the saloon, causing the party to officially commence:<blockquote>At the top of the stairway stood the Duchess of Devonshire, attired with great Oriental magnificence as Zenobia, to receive her guests. Her dress was tissue of silver in front, wrought with jewels. The over-dress was cloth of gold, magnificently wrought with jewels, and her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls. The whole front of the bodice was hung with jewels, and strings of pearls were worn round the neck.<ref>“Historic Ball at Devonshire House. Brilliant Scene.” The ''Daily Telegraph'' 3 July 1897, Saturday: 9 [of 14], Col. 6a–7c [of 7]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18970703/094/0009.</ref>{{rp|9, Col. 6a}}</blockquote> * According to the London ''Echo'',<blockquote>THE DUCHESS of DEVONSHIRE was a dazzling vision, dressed as “Zenobia,” in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems, outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe [may be crêpe] de chine, delicately worked in crystals and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The corsage was to correspond, and had a magnificent girdle of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin, and about the head — one magnificent pear-shaped pearl resting on the forehead. Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail.<ref>“A Jubilee Ball. Brilliant Scene at Devonshire House. Some of the Costumes Worn.” The London ''Echo'' 3 July 1897, Saturday: 2 [of 4], Cols. 6a – 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004596/18970703/027/0002.</ref>{{rp|2, Col. 6c}}</blockquote> * The report on the ball in ''Truth'' is written as a letter from one girl to another, named Amy: "In tissue of silver and cloth of gold, and richly jewelled from head to foot, stood the stately Zenobia, Duchess of Devonshire, at the head of her marble stairway, to receive her guests of all the ages."<ref name=":29">“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.</ref>{{rp|41, Col. 1b}} The article ends back in the voice of the gossiping girl: "I have no space to tell you of any more, but you may possibly gather some idea of the splendour of the ball from these scanty notes about the gowns and jewels."<ref name=":29" />{{rp|42, Col. 2c}} * According to the article in ''The Graphic'' written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], although this caption to the Lafayette photograph may not be hers, the Duchess of Devonshire wore a<blockquote>Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds.<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}}</blockquote> * According to ''The Gentlewoman'', a newspaper that often covered fashion and design,<blockquote>The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees.<ref name=":22" />{{rp|p. 32, Col. 1c–2a}}</blockquote> * This description from the ''Queen'' accompanies a flattering line drawing of the Duchess of Devonshire in costume (Bottom left drawing, from the back, emphasizing the train. Initials for artist, illegible right now.]:<blockquote>D<small>RESS</small> worn by the D<small>UCHESS OF</small> D<small>EVONSHIRE</small>, as Zenobia, Queen of Palmyra. — The skirt of gold tissue, embroidered all over in a star-like design in emeralds, sapphires, and diamonds, and other jewels outlined in gold; the corners where it opens in front are elaborately wrought in the same jewels and gold to represent peacocks’ outspread tails. This opened to show an under-dress of cream crêpe de Chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train is attached by two slender points to the shoulders, and is of green velvet of a lovely shade, embroidered in an Oriental design in rubies, sapphires, amethysts, emeralds, and diamonds. Headdress of real diamonds, with large emeralds and rubies, and two ostrich feathers in front, with festoons of pearls.<ref>“Dress at Devonshire House on July 2.” The ''Queen, The Lady’s Newspaper'' 10 July 1897, Saturday: 39 [of 98 in BNA; p. 65 on print page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/137/0039.</ref>{{rp|p. 39, Col. 1b–c}}</blockquote> === Observations on the Duchess's Costume === Because it is in black and white, the photograph of the Duchess as Zenobia taken by Lafayette (above right) can tell us almost nothing about the colors of the costume. Fortunately, this costume has survived and is held in Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth). ''House Style: Five Centuries of Fashion at Chatsworth'', the volume about an exhibition mounted 25 March to 22 October 2017, has beautiful photographs from several perspectives of the costume, including the train, and accessories.<ref name=":18" /> Photographs from this exhibition have appeared in a number of places on the web and social media. A few online magazines have printed images, apparently from this exhibition, as well. This costume was worn by Deborah, 11th Duchess of Devonshire when the Dowager Duchess in 2000 for her 80th birthday.<ref name=":25">@saadsalman719. {{Cite web|url=https://royalwatcherblog.com/2016/09/24/the-tiaras-of-debo-duchess-of-devonshire/|title=Duchess of Devonshire’s Tiaras|last=Saad719|date=2016-09-24|website=The Royal Watcher|language=en-us|access-date=2022-12-04}} https://royalwatcherblog.com/2016/09/24/the-tiaras-of-debo-duchess-of-devonshire/.</ref> She wore it with the Devonshire Palmette tiara, which also was made for the 8th Duchess, in 1893, shortly after she married the 8th Duke.<ref name=":25" /> The 3-row base of this 1900-diamond tiara was added in 1897,<ref>{{Cite web|url=http://orderofsplendor.blogspot.com/2014/09/tiara-thursday-devonshire-diamond-tiara.html|title=The Royal Order of Sartorial Splendor: Tiara Thursday: The Devonshire Diamond Tiara|date=2014-09-25|website=The Royal Order of Sartorial Splendor|access-date=2022-12-04}} http://orderofsplendor.blogspot.com/2014/09/tiara-thursday-devonshire-diamond-tiara.html.</ref> but Louise, the 8th Duchess, did not wear the Palmette tiara to the ball; [[Social Victorians/People/Louisa Montagu Cavendish#Zenobia's Crown|another headdress]] was made for her costume. In the photograph of her in costume, Louise, the 8th Duchess is standing in a three-quarter front pose. That static position in a two-dimensional image makes it difficult to understand exactly how the costume and accessories worked. The terms we use are these: [[Social Victorians/Terminology#Train|train]], [[Social Victorians/Terminology#Bodice|bodice]] (what's on the torso, no matter how many layers), [[Social Victorians/Terminology#Stomacher|stomacher]], belt (really an accessory, like the jewelry), sleeves, overskirt, [[Social Victorians/Terminology#Petticoat|petticoat]] (or underskirt) and [[Social Victorians/Terminology#Frou-frou|frou-frou]]. Some of the 1897 terminology for parts of the dress differs from what we would use now. Some of the terms that appear in the newspaper reports include front, bodice, [[Social Victorians/Terminology#Corsage|corsage]], stomacher, [[Social Victorians/Terminology#Girdle|girdle]], train and overmantle, skirt, gown, robe and under-dress. ==== Inconsistencies in the Descriptions and Images ==== * The newspaper accounts of the Duchess's costume do not agree on several particulars, including the color of the train and its lining. * Inconsistencies in the photographs continue into the 21st century. We see that the fit is different on the body of Deborah, 11th Duchess of Devonshire in 2000 and on the mannequin at the 2017 exhibition. Most striking is the color of [[Social Victorians/People/Louisa Montagu Cavendish#The Original Train|the train]] of the dress, which ranges in the several recent photographs available online from a "primary" green to a goldish green, possibly in part due to the lighting, background and quality of the print of the photograph. Possibly, also, of course, the color may be fading and changing as the fabric is exposed to light. * The dress is loosely draped on the body of Louise, 8th Duchess and on Deborah, 11th Duchess. It appears stiffer and drapes less gracefully in the 2017 exhibition photographs. ==== The Duchess's Body ==== The costume appears to have been designed to flatter the 8th Duchess's body. * She was 65 years old at the time of the ball. * She was unusually beautiful in her youth. In her 1984 ''The Duchess of Devonshire's Ball'', Sophia Murphy says, "As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} * She was confident, a "political hostess," "said to be the most powerful person outside the government."<ref name=":20" />{{rp|12}} * [[Social Victorians/People/Marlborough|Consuelo, Duchess of Marlborough]] describing Louise, Duchess of Devonshire, says, "Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash ...."<ref name=":23" />{{rp|115}} She said this a few months after Queen Victoria's death in 1901, but they knew each other in 1897: the Duchess of Marlborough was at the ball. * Although it is impossible to tell anything about what undergarments she may have been wearing from the Lafayette portrait, the Duchess does not appear to have a tightly laced waist. The drape of the costume suggests it might have been intended to be more comfortable than to show off her waist. * Because her almost-straight arms appear to be deliberately (and perhaps unnaturally) positioned over the front of her body, she looks as if she might be self-conscious. * Louise typically wore earrings in portraits other than Lafayette's and appears to have had pierced ears. The costume hangs differently on Deborah, the 11th Duchess of Devonshire in photographs of her in 2000, suggesting that it was altered to fit her body. It also hangs differently on the mannequin used in the 2017 Devonshire House Style exhibition, in part because the bottom of the skirts of the petticoat and overdress were extended to showcase the decorations at the hem. ==== Fit ==== *The recent exhibitions of the dress show a different fit than what is shown in the Lafayette photograph. Louise, the 8th Duchess's bodice is draped loosely over the stomacher, but photographs of Deborah, 11th Duchess or of the mannequin in the 2017 exhibition show a bodice fitted to the stomacher. * Also, the costume itself may have been altered or adjusted since 1897 to make the waist more defined and the line more Elizabethan (the wide, square neckline; the A-line silhouette of the skirt; the split in the over-skirt with the petticoat or under-skirt in the middle). * The dress seems to have been designed to flatter the 8th Duchess's body, especially her waist: the belt falls down to the hips; the skirt has an A-line shape; what we can see of the under-dress is narrow at the top and wider towards the bottom. * The drape of the skirt falls in a straighter line on the 8th Duchess as well as the Dowager Duchess than on the mannequins. In the exhibition, the bottom of the skirt on the mannequin has been extended outward, accentuating the A-line, in order to display the complex embroidery and ornamentation. The encrustation of the jewels and embroidery would stiffen the bottom of the skirt, but its weight would pull the skirt down; it would not naturally spread out in the way it does on the mannequin. ==== The Original Dress (Minus the Train) ==== The duchess's dress was made to look like an underdress and an overdress, although of course the construction was actually quite a bit more complex than this and includes more parts. The layered dress has a bodice made up of a [[Social Victorians/Terminology#Stomacher|stomacher]] and a [[Social Victorians/Terminology#Corsage|corsage]], with a [[Social Victorians/Terminology#Petticoat|petticoat]] and skirt over it. The garments are made to be a single costume. This Worth dress is covered in typically late-19th-century [[Social Victorians/Terminology#Frou-frou|frou-frou]]. * The stomacher is fitted to the body, while the corsage (which, with the overskirt, is made to look like a dress) hangs more loosely from the shoulders to the waist. Similarly, the embroidered patterns on the stomacher and underskirt match, suggesting that they too are of a piece. * According to the Pall Mall ''Gazette'', "One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy."<ref name=":24">“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref> * The stomacher and petticoat are look like an underdress in part because the fabric and trim match. The trim gives a three-dimensional pattern on the fabrics as well a depth and richness to the edgings. For example, the pattern on the fabric of the underdress appears to be beaded all over as well as embroidered. The pieces of the overdress also match in fabric and trim. * The 8th Duchess did not take part in the Artistic Dress fashion of not wearing a corset, although her costume does not show the very small waist some women at the ball achieved with tight lacing. In the Lafayette portrait, Louise's stomacher appears to be fitted to her body while the overdress is more loosely draped. * In the 1897 Lafayette portrait, the Duchess's petticoat, or underskirt, appears to be A-line, although it hangs straight down because of the weight of the embroidery at the bottom edge. * For the 2017 exhibition, the overdress and the petticoat were fully extended at the bottom to reveal the A-line drape of the skirts and to showcase the artistry and complexity of the trim. * Like the petticoat, the overskirt appears to be A-line. The heavy embroidery at the bottom of the overskirt is completed visually by the design at the bottom of the underskirt. * Louise's [[Social Victorians/Terminology#Décolletage|décolletage]] is not as revealing as was fashionable and acceptable at this time, suggesting perhaps a modesty in dress, but baring her upper arms, which is perhaps more unusual, makes this costume more daring than it would otherwise be. * The unusual sleeves appear to have no seam under the arm. They are open on the outside of the arm and fall below the waist on the under-arm side, and the two edges are attached at the top of the shoulders, to simulate flower petals. * The fabric of the sleeves appears to be sheer, different from any other fabric in the dress. An embroidered and beaded paisley pattern — only very tangentially related to the shapes in the rest of the trim — is stitched along the edges. * In the 1897 photograph, the Duchess's arms are quite bare, perhaps in part as a result of their position. In the 2000 photograph of the Dowager Duchess and the 2017 mannequins, the sleeves show the petal-like structure. In the later photographs, the dress is draped differently, showing less of the upper arm. In the photographs of the 11th Duchess, the dress appeared to have been altered and the top of the sleeves appear to be overlapped instead of caught up. * The sequins used to embellish the costume appear to be silver, held in place with a bead sewn at the center. [[Social Victorians/Victorian Things#Sequins and Spangles|Sequins]] themselves have a long history and were probably mass-produced at the time of the ball. The silver ones like the ones used in the Duchess's costume were useful in garments worn only once because they would have tarnished, turning black and dull. * This costume would have been worn only once by the 8th Duchess not only because the silver work would have tarnished but also because it was temporarily decorated by real stones and pearls from her jewelry.<ref name=":24" /> <ref name=":26" /> * The thread and cords used for the trim were made out of real silver and gold filaments. ==== The Original Train ==== * While the [[Social Victorians/Terminology#Train|train]] looks like it falls from the shoulders, its heaviness would have distorted the dress, pulling the shoulders down in the back. It may have been made to be a part of the corsage and attached at the waist as well as at the shoulders, transferring more of the weight to the waist. The weight of a heavy garment like this train was a problem Victorian designers faced often, usually by spreading the weight of the garment to the wearer's waist. The London ''Evening Standard'', ''Morning Post'', ''Times'' and ''Daily News'' all report that the train was "fastened" or "caught" at the waist with a diamond ornament or brooch, but this attachment to the dress was likely far more structural and less decorative. * The color of the green velvet on the train differs depending on the photograph. One photograph from the German ''Royal Magazine'' shows a very vivid green,<ref>{{Cite web|url=https://royal-magazin.de/england/devonshire/zanobia-costume-headrdress-loise-duchess-of-devonshire.htm|title=Zenobia Costume and Headdress Louise Duchess of Devonshire{{!}} Double Duchess{{!}}Royal Noble Jewel History|website=royal-magazin.de|access-date=2022-10-20}} https://royal-magazin.de/england/devonshire/zanobia-costume-headrdress-loise-duchess-of-devonshire.htm.</ref> although it is possible the photograph enhanced the greenness of the train to emphasize the embroidery on it. One photograph, used in two blogs without attribution, shows a yellowish color to the train.<ref>{{Cite web|url=https://advantageinvintage.co.uk/2012/01/20/the-duchess-of-devonshires-ball-1897/|title=The Duchess of Devonshire’s ball 1897|last=liztregenza|date=2012-01-20|website=Advantage In Vintage|language=en|access-date=2022-10-20}}</ref><ref>{{Cite web|url=https://oldrags.tumblr.com/post/35083772903/louise-duchess-of-devonshires-queen-of-zenobia|title=Old Rags|last=oldrags|website=Tumblr|access-date=2022-10-20}}</ref> Photographs from Chatsworth House might be trusted to have truer color; published in the Chatsworth House newsletter, they are no longer directly accessible. * Other photographs show a lighter green. This change in the color needs exploration. Without looking at the actual garment, it is impossible to know why the color appears to have changed in the several photographs. The causes could range from the condition of the fabric or the dye to the lighting of the garment during the photographing process. Furthermore, the dye formula (and mordant, used to set the dye) could be very interesting to research (although green dyes containing arsenic were no longer in use by the 1890s<ref>"Scheele's Green." ''Wikipedia'' https://en.wikipedia.org/wiki/Scheele%27s_Green (retrieved 17 October 2022).</ref>). * The reporter's description of the Duchess's train in the London ''Daily News'' is confusing: it says the train is "bright green velvet" and "It was a mass of gorgeous embroidery, carried out in heliotrope velvet." Embroidery is not "carried out in" velvet. Heliotrope is not green, it's pinkish purple. * Heliotrope was not green in 1897, either: heliotrope appears to have been a new word for a color, introduced perhaps in 1882 in A. Maerz and M. Rea Paul's ''A Dictionary of Colour''.<ref>A. Maerz and M. Rea Paul. "Color Sample of Heliotrope: Plate 54 Colour Sample C10." ''A Dictionary of Colour''. McGraw-Hill: 1930: 131.</ref><ref>"Heliotrope," ''Wikipedia'' https://en.wikipedia.org/wiki/Heliotrope_(color)#cite_note-2 (accessed November 2022).</ref> The ''OED'' lists 3 newspaper sources for the 1880s, The ''World'' (1882), ''Truth'' (1886), and the ''Daily News'' (1887).<ref>"heliotrope, n.". ''OED Online''. September 2022. Oxford University Press. https://www-oed-com.scsuproxy.mnpals.net/view/Entry/85617?redirectedFrom=heliotrope (accessed August 2022).</ref> The ''Daily News'' describes it we<nowiki/>ll: "that peculiar mauve known as heliotrope." The word itself has existed in English for many centuries, both for the flower (which turns to follow the sun) as well as for instruments that reflect or measure sunlight and also for a variety of quartz (also called bloodstone). ==== Mixing the Present and the Past ==== Historical fiction — whether novels, paintings or costumes — is always about both the past and the present moment. The costume that Louise, the 8th Duchess wore was designed by Attillo Comelli and constructed by the House of Worth. Comelli also designed the costumes of the fan bearers, trumpeters and children in her entourage. The historical Zenobia's Palmyrene Empire was Syrian, and the entourage's costumes were described as Syrian (Assyrian)<ref name=":5" /> or "Babylonish." The ''Westminster Review'' says that the Duchess of Devonshire took part in the "Oriental" procession, but no other newspaper, including the ''Morning Post'' and the ''Times'', which printed details about the processions, include her.<ref name=":9" /> The processions were a means of formally presenting the guests to the Prince and Princess of Wales. The Duchess would not have needed to be presented to the royals: she and the Duke welcomed them to Devonshire House when they first arrived. The fact that Comelli designed the Duchess's costume and the House of Worth constructed it is important because the House of Worth designed fashion. Even though this was a fancy-dress ball to which they were expected to wear costumes, the guests would always have given fashionably flattering clothing precedence over what we might now call historical accuracy. Even though Comelli would have been familiar with early clothing history, it is unlikely he would have been a stickler for "historical accuracy." He would have been accustomed to designing costumes that took into account the body (if not always the taste) of the wearer. That is, while he was likely the source of what historical accuracy existed in the costume, his designs were always Victorian treatments of the historical. * In the 1897 portrait, the sleeve treatment and the general drape of the bodice and the skirt are consistent with what we think Victorians would consider ancient Syrian. We see the hand of Comelli, in particular, in the way the bodice and skirt are layered and not tightly fitted (possibly for comfort), the symbols and shapes that make up the trim and the way the skirt falls straight down from the hips. * Perhaps the Duchess's bare arms were meant to be historical and suggest the climate of Palmyra. * The costume is not "historically accurate" to 3rd century Palmyra, however, in any way. Beyond Comelli's likely treatment of the historical are embellishments common in formal clothing of the most wealthy and fashionable at the end of the 19th century, especially in the way the [[Social Victorians/Victorian Things#Sequins and Spangles|sequins]] and the shaped and faceted precious and semi-precious stones were used in the trim. * The versions of the costume shown in recent exhibitions make it seem more Victorian than what we see in Lafayette's portrait of the 8th Duchess, especially the waist and the fit of the bodice. * What makes the costume look so very typical of the late 19th century are its general flow or silhouette, the lines of the bodice and skirt, the fitted waist, the fit of the dress against the stomacher and the belt. ** The flow of the costume, which centers at the waist, has a Victorian shape. The two ends of the belt bring together the point of the stomacher and the lines made by the edges of the overdress, making this center. ** The lines of the bodice and skirt are Victorian in the fitted waist and the A-line of the skirt. They appear more Victorian in the latter photographs, which accentuate the A-line, than they do in the 1897 portrait. ** The belt (discussed generally in [[Social Victorians/People/Louisa Montagu Cavendish#Accessories|Accessories]], below) looks very Victorian. ** In the more recent photographs, the tightness of the dress against the stomacher all look Victorian. * As a piece that looks like it is attached to rather than part of the dress, the train has an ahistorical quality, not really belonging to the ancient Syrians nor what was common to late-Victorian ballgowns. In a way, its purpose can be said to have been merely decorative, to showcase the embroidery and encrustations of jewels, pearls and sequins. * Similarities between late-Victorian and late-16th-century styles, which we can see in the Lafayette portrait, complexify the elements that are not ancient Syrian: the square neckline, the A-line shape of the skirts and the focal point just below the waist. ==== Trim ==== * The trim consists of complex matching patterns made up of cords, jewels, pearls, beads, sequins and embroidery. * The jewels are framed, likely by gold cording, and put into settings, which makes the jewels pop — emphasizing them and giving them three-dimensionality. * The beading at the edges of the garment pieces adds weight and helps manage the drape of the fabric. * The sequins attached to the costume would have been metal. Since silver tarnishes over time — and tarnish cannot be removed without damage to the fabric — the sequins, beads and embroidery thread that are dark now may be silver. * The gold thread in the embroidery has not tarnished, but the silver thread, which would have sparkled in 1897, has. ==== Accessories ==== In the 1897 Lafayette portrait of her in her costume, the Duchess also has some accessories and props. Not shown in the photograph is her entourage, perhaps 6 people in total, who also had costumes, accessories and props. In her portrait, we see Louise with a complex and ornate headdress, a fan, quite a bit of jewelry, and perhaps a handkerchief tucked into one bracelet. * The belt is an elaborate accessory to the costume and heavily encrusted with sequins, beads and jewels, its weight and stiffness holding it in place. The focal point of the costume is the front center, near the waist, where the belt, the stomacher, the overdress and the petticoat all come together. This focal point is defined clearly by the meeting of the highly decorated two ends of the belt. * Mostly hidden in the 1897 Lafayette portrait, the belt falls perhaps very slightly below the Duchess's waist. On the 80-year-old 11th Duchess, it appears to be closer to her waistline instead of falling below it. On the mannequin in the 2017 exhibition, the belt clearly falls below the waist in front. * Based on what the Duchess of Marlborough said in 1901 and on how dark the hair is in Lafayette's portrait (she's 65), Louise is likely wearing a wig here. Because of the angle of her head in the portrait and the elaborate headdress, it is impossible to tell whether her hair is up or down or a little of both. Because of the angle of her body, the sleeve treatment and the fact that the photograph is black and white, the dark shape behind her might be a long braid falling down the center of her back. * The fan carried by the Duchess, at least in the Lafayette photograph, appears to be made of long white ostrich plumes. It would have had a handle, and in this case, likely a highly ornate one. * Louise's complex headdress looks like [[Social Victorians/People/Louisa Montagu Cavendish#Zenobia's Crown|a crown]], with a jewel-encrusted band, crescent-shaped horns, loops of pearls, a large tear-drop pearl in the center of the forehead, and white feathers (likely the traditional 3 plumes worn for the Prince of Wales). * Louise's jewelry is addressed in [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|its own section]]. * Can't tell how many strands of pearls are around Louise's neck in the Lafayette photo, but the photographs of the recreation of the crown and of the costume from the 2017 exhibit show [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|five strands]]. * The Duchess is wearing bracelets on both wrists in the Lafayette photograph. The most visible one — on her left arm — appears to be a simple thick band, with a handkerchief tucked under it. == The Duchess's Jewelry == Newspaper reports before the ball gossiped about the jewelry associated with the costumes being developed. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref name=":26">“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> The Pall Mall ''Gazette'' says, "One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy."<ref name=":24" /> (The [[Social Victorians/Terminology#Stomacher|stomacher]] seems less heavily encrusted with jewels than the belt or the part of the bodice that looks as if it is part of the overdress, so this reporter may have made a mistake, or perhaps this writing is simply imprecise in general, as ''form a stomacher'' is also really not very clear.) The Duchess was known for her jewelry, and as she aged and after her marriage to the 8th Duke in 1892, the jewelry she wore only increased in scale and notability. It is not surprising that her jewelry would provide copy for the newspapers. The jewelry she wore at the ball consisted of a headdress that included loops of pearls, pearls around her neck, bracelets and at least one ring. (She does not seem to be wearing earrings in the Lafayette portrait.) Besides the jewels and pearls Louise wore as jewelry to the ball, the embroidery on every piece of her costume was "encrusted" with jewels, pearls, beads and sequins. Some accounts of her costume suggest that all the jewels sewn onto the dress were actual precious or semiprecious stones, but the London ''Daily News'' says some of them were crystals.<ref name=":15" />(p. 5, Col. 6a) According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>The skirt of gold gauze, appliquéd with tinsel medallions and peacock plumes worked in bright foils, wire coils and spangled with sequins, was worn over an ivory satin underskirt wrought over with silver thread and diamonds. Attached to the shoulders was a long graduated train in the most vivid emerald-green velvet, appliquéd with velvet and gold work in an eastern design and studied with jewels.<ref name=":18" />(128)</blockquote> The encrustation of embroidery, jewels and baubles would have made the costume heavy, and concentrating all this [[Social Victorians/Terminology#Frou-frou|frou-frou]] at the bottom edges would have affected the drape and fold of not only the skirts and train but the stomacher, bodice and sleeves. The density of the decoration on the train and overskirt caused the drape to remain open and flat rather than letting the fabric fold. The structural foundation provided by the "petticoat," or underskirt, which likely continued completely around her body, also would have helped prevent folding. Because they would have been made of metal, the sequins alone would have added considerably to the weight of the costume. Because they are so dull and black in the recent color photographs, the sequins are probably made of silver and have tarnished over time. The dull black beads are also probably silver. Some of the embroidery contained metal threads made of silver or gold. While the gold would not tarnish, the silver has blackened. Removing this tarnish is impossible using today's methods because of how small the sequins, beads and thread are and because of the damage it might do to the fabrics. Other beads might have been made of semiprecious stones, pearls or glass, which would also have weighed down the fabric. === Newspaper Reports about the Jewels === Almost all newspaper reports of her costume at the ball mention her jewels because they were so much a topic of conversation. Because they were sewn onto the costume itself, these descriptions emphasize her jewelry. Most are very similar to each other: * The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref name=":24" />{{rp|p. 7, Col. 2b}} * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees." <ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} <ref name=":22">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} As with the colors, Lafayette's photograph of the Duchess in costume does not show the jewels very clearly. We cannot see the stomacher or the "long chains of pearls and other wonderful jewels" or the pearls that "hung down almost to the knees" (although, of course, because reporters were not present at the ball, any newspaper account can be wrong). === Zenobia's Crown<ref name=":10" /> === The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right), and no other images of the crown exist. It was lavish, "encrusted" with jewels and featuring pearls: * London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />{{rp|p. 5, Col. 6a}} * These descriptions are all identical except for the addition or subtraction of an occasional comma: The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6" />{{rp|p. 12, Col. 3b}} <ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} Do the horns on her headdress go toward the back from the apex that rises from the jewel-encrusted band or do they stick straight out to the sides? Does the headdress include the traditional three white plumes? This crown did not survive: it was recreated from the single Lafayette photograph of the Duchess in costume for the exhibition mounted in 2017, ''House Style: Five Centuries of Fashion at Chatsworth'', and exhibited on the mannequin wearing the Duchess's costume. A photograph of this recreation separate from the costume, but with the pearls and the white plumes, can be seen on [https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery this Sotheby's page] about C. W Sellor's, the jewelry firm that did the recreation as well as a number of other recreations for the exhibit.<ref>Sotheby's. "How C. W. Sellors Transformed Five Centuries of Fashion Into Modern Jewellery." ''Sotheby's'' 9 May 2017. Retrieved 11 August 2022. https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery.</ref> Although Lafayette's photograph is not conclusive, in Sellor's recreation the horns on the crown point toward back of the head rather than out to the sides. === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts independent of the general commentary on her jewelry. The Lafayette portrait does not make clear how many strands of pearls the Duchess's necklace contained, but it is likely 5 strands. In later appearances of the costume (on Deborah, 11the Duchess and on the mannequin in 2017), the necklace has five strands, but of course a pearl necklace strung at the end of the 19th century will have been restrung by the 21st century (assuming that it is even the same necklace). Since at least 1892, Louise, Duchess of Devonshire had had a pearl "necklet" with five strands, which was restrung on 1 March 1897. The invoice and receipt of this restringing in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some of the Duchess's pearls. The restringings appear on the same invoice, the first occurring apparently on 15 October 1892 and the invoice sent on 20 October 1892. Possibly relevant to the Duchess's costume is a restringing dated 1 March 1897. The first necklace is a "Pearl Necklet in original 4 rows." The invoice is dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>{{rp|1}} ==== The Original Necklet: 4 Rows ==== The necklet she brought in to be restrung contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls ==== 1892 Restringing: 5 Rows ==== The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small"<ref name=":14" />{{rp|1}}: # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small ==== 1897 Restringing: 5 Rows ==== The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />{{rp|2}}: # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a coronet, but the handwriting is not clear enough to read.<ref name=":14" />{{rp|2}} If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the third necklet, strung in 1897 the spring before the ball, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Historical Zenobia == Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire — at the edge of the Roman empire — ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> The historical Zenobia as well as fictional characters merely named Zenobia were popular subjects in 19th century art, literature (including Benjamin Disraeli, Nathaniel Hawthorne and Walter Savage Landor), opera, sculpture, and paintings. She was also the subject of travel books. Middle-eastern traveller Lady Hester Stanhope (1776–1839), for example, discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> We do not know what sources, if any, were consulted by Louise, Duchess of Devonshire in preparing for personating Zenobia for the ball. Besides what information might have been provided at museum and gallery exhibits, non-scholarly Victorians had available to them Edward Gibbon's ''History of the Decline and Fall of the Roman Empire'' and the ''Encyclopædia Britannica'' as well as occasional books or articles in the more substantial magazines. Noting, as always, language that is offensive to us now, Edward Gibbon's 1761–1789<ref>{{Cite journal|date=2022-10-30|title=The History of the Decline and Fall of the Roman Empire|url=https://en.wikipedia.org/w/index.php?title=The_History_of_the_Decline_and_Fall_of_the_Roman_Empire&oldid=1119141639|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/The_History_of_the_Decline_and_Fall_of_the_Roman_Empire.</ref> ''History of the Decline and Fall of the Roman Empire'' includes Zenobia: <blockquote>Modern Europe has produced several illustrious women who have sustained with glory the weight of empire; nor is our own age destitute of such distinguished characters. But if we except the doubtful achievements of Semiramis, Zenobia is perhaps the only female whose superior genius broke through the servile indolence imposed on her sex by the climate and manners of Asia. She claimed her descent [19–20] from the Macedonian kings of Egypt, equalled in beauty her ancestor Cleopatra, and far surpassed that princess in chastity and valour. Zenobia was esteemed the most lovely as well as the most heroic of her sex. She was of a dark complexion (for in speaking of a lady these trifles become important). Her teeth were of a pearly whiteness, and her large black eyes sparkled with uncommon fire, tempered by the most attractive sweetness. Her voice was strong and harmonious. Her manly understanding was strengthened and adorned by study. She was not ignorant of the Latin tongue, but possessed in equal perfection the Greek, the Syriac, and the Egyptian languages. She had drawn up for her own use an epitome of oriental history, and familiarly compared the beauties of Homer and Plato under the tuition of the sublime Longinus. This accomplished woman gave her hand to Odenathus, who, from a private station, raised himself to the dominion of the East. She soon became the friend and companion of a hero. In the intervals of war Odenathus passionately delighted in the exercise of hunting; he pursued with ardour the wild beasts of the desert, lions, panthers, and bears; and the ardour of Zenobia in that dangerous amusement was not inferior to his own. She had inured her constitution to fatigue, disdained the use of a covered carriage, generally appeared on horseback in a military habit, and sometimes marched several miles on foot at the head of the troops. The success of Odenathus was in a great measure ascribed to her incomparable prudence and fortitude. Their splendid victories over the Great King, whom they twice pursued as far as the gates of Ctesiphon, laid the foundations of their united fame and power. The armies which they commanded, and the provinces which they had saved, acknowledged not any other sovereigns than their invincible chiefs. The senate and people of Rome revered a stranger who had avenged their captive emperor, and even the insensible son of Valerian accepted Odenathus for his legitimate colleague. After a successful expedition against the Gothic plunderers of Asia, the Palmyrenian prince returned to the city of Emesa in Syria. Invincible in war, he was there cut off by domestic treason, and his favourite amusement of hunting was the cause, or at least the occasion, of his death. His nephew, Mæonius, [p. 20 / p. 21] presumed to dart his javelin before that of his uncle; and, though admonished of his error, repeated the same insolence. As a monarch, and as a sportsman, Odenathus was provoked, took away his horse, a mark of ignominy among the barbarians, and chastised the rash youth by a short confinement. The offence was soon forgot, but the punishment was remembered; and Mæonius, with a few daring associates, assassinated his uncle in the midst of a great entertainment. Herod, the son of Odenathus, though not of Zenobia, a young man of a soft and effeminate temper, was killed with his father. But Mæonius obtained only the pleasure of revenge by this bloody deed. He had scarcely time to assume the title of Augustus before he was sacrificed by Zenobia to the memory of her husband. With the assistance of his most faithful friends, she immediately filled the vacant throne, and governed with manly counsels Palmyra, Syria, and the East, above five years. By the death of Odenathus, that authority was at an end which the Egypt senate had granted him only as a personal distinction; but his martial widow, disdaining both the senate and Gallienus, obliged one of the Roman generals who was sent against her to retreat into Europe, with the loss of his army and his reputation. Instead of the little passions which so frequently perplex a female reign, the steady administration of Zenobia was guided by the most judicious maxims of policy. If it was expedient to pardon, she could calm her resentment; if it was necessary to punish, she could impose silence on the voice of pity. Her strict economy was accused of avarice; yet on every proper occasion she appeared magnificent and liberal. The neighbouring states of Arabia, Armenia, and Persia, dreaded her enmity, and solicited her alliance. To the dominions of Odenathus, which extended from the Euphrates to the frontiers of Bithynia, his widow added the inheritance of her ancestors, the populous and fertile kingdom of Egypt. The emperor Claudius acknowledged [p. 21 / p. 22] her merit, and was content that, while he pursued the Gothic war, ''she'' should assert the dignity of the empire in the East. The conduct, however, of Zenobia was attended with some ambiguity; nor is it unlikely that she had conceived the design of erecting an independent and hostile monarchy. She blended with the popular manners of Roman princes the stately pomp of the courts of Asia, and exacted from her subjects the same adoration that was paid to the successors of Cyrus. She bestowed on her three sons a Latin education, and often showed them to the troops adorned with the Imperial purple. For herself she reserved the diadem, with the splendid but doubtful title of Queen of the East. When Aurelian passed over into Asia, against an adversary whose sex alone could render her an object of contempt, his presence restored obedience to the province of Bithynia, already shaken by the arms and intrigues of Zenobia. ... Antioch was deserted on his approach, till the emperor, by his salutary edicts, recalled the fugitives, and granted a general pardon to all who, from necessity rather than choice, had been engaged in the service of the Palmyrenian queen. The unexpected mildness of such a conduct reconciled the minds of the Syrians, and, as far as the gates of Emesa, the wishes of the people seconded the terror of his arms. Zenobia would have ill deserved her reputation, had she indolently permitted the emperor of the West to approach within an hundred miles of her capital. The fate of the East was decided in two great battles; so similar in almost every circumstance, that we can scarcely distinguish them from each other, except by observing that the first was fought [p. 22 / p. 23] near Antioch, and the second near Emesa. In both the queen of Palmyra animated the armies by her presence, and devolved the execution of her orders on Zabdas, who had already signalised his military talents by the conquest of Egypt. The numerous forces of Zenobia consisted for the most part of light archers, and of heavy cavalry clothed in complete steel. The Moorish and Illyrian horse of Aurelian were unable to sustain the ponderous charge of their antagonists. They fled in real or affected disorder, engaged the Palmyrenians in a laborious pursuit, harassed them by a desultory combat, and at length discomfited this impenetrable but unwieldy body of cavalry. The light infantry, in the mean time, when they had exhausted their quivers, remaining without protection against a closer onset, exposed their naked sides to the swords of the legions. ... After the defeat of Emesa, Zenobia found it impossible to collect a third army. As far as the frontier of Egypt, the nations subject to her empire had joined the standard of the conqueror, who detached Probus, the bravest of his generals, to possess himself of the Egyptian provinces. Palmyra was the last resource of the widow of Odenathus. She retired within the walls of her capital, made every preparation for a vigorous resistance, and declared, with the intrepidity of a heroine, that the last moment of her reign and of her life should be the same.<ref>{{Cite book|url=https://books.google.com/books?id=Bd_LZYcY3UIC&pg=PA19&dq=Zenobia&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwir38Kjudj5AhX7kokEHW2BDF04KBDoAXoECAIQAg#v=onepage&q=Zenobia&f=false|title=The History of the Decline and Fall of the Roman Empire|last=Gibbon|first=Edward|date=1887|publisher=J. Murray|language=en}} Retrieved 11 August 2022. https://books.google.com/books?id=Bd_LZYcY3UIC. Footnotes and similar annotations silently elided.</ref>{{rp|Ch. XI, 19–23}} </blockquote> The ''Encyclopædia Britannica'' offers a sense of what was generally available to those interested in that kind of research. The discussion of Zenobia that would have been available in 1897, in the 9th edition of the ''Britannica'', is embedded in the article on Palmyra. It begins with her husband, Odænathus, and his father: <blockquote> Odænathus himself seems to have been engaged in almost constant warfare in the east and north against the Persians and perhaps the Scythians, but in his absence the reins of government were firmly held by his wife Zenobia, the most famous heroine of antiquity, to whom indeed Aurelian, in a letter preserved by Trebellius Pollio, ascribes the chief merit of all her husband's success. Septimia Zenobia was by birth a Palmyrene; her native name was Bath Zabbai ... [;] and Pollio's description of her dark beauty, black flashing eyes, and pearly teeth, together with her unusual physical endurance and the frank commanding manners which secured her authority in the camp and the desert, point emphatically to an Arabic rather than a Syrian descent. ... To the union of firmness and clemency, which is the most necessary quality of an Eastern sovereign, Zenobia added the rarer gifts of economy and organization, and an unusual range of intellectual culture. She spoke Coptic as well as Syriac, knew something of Latin, and had learned Greek from the famous Longinus, who remained at her court to the last, and paid the penalty of his life for his share in her counsels. She was also a diligent student of Eastern and Western history, and the statement that she enjoined her sons to speak Latin so that they had difficulty in using Greek implies a consistent and early adoption of the policy which made the sucess of Odænathus, and, taken in connexion with Aurelians' testimony, in a letter preserved by Pollio, that she had the chief merit of her husband's exploits, seems to justify the conclusion that it was her educated political insight that created the fortunes of the short-lived dynasty. ... In the zenith of his fame Odænathus was cut off by assassination along with his eldest son Herod, and it is generally assumed that the murder took place under Gallienus. ... [p. 201, Col. 2c – p. 202, Col. 1a] The fact seems to be that, while Odaenathus was busy at the other end of his kingdom, Zenobia administered the government at Palmyra and directed the conquest of Egypt, still nominally acting under the emperor at Rome, whose authority on the Nile was disputed by one or more pretenders. ... It still seems ... strange and yet an undoubted fact that Zenobia, who not only enjoyed the real authority behind her beardless son, but placed her name before his on public inscriptions, ... struck no coins till the second year of Aurelian, when the breach with Rome took place, and she suddenly appears as an empress (Σεβαστή, Augusta) of five years' standing. Up to that date the royal pair probably did not venture to coin in open defiance to Rome, and yet were unwilling to circulate an acknowledgment of vassalship in all the bazaars of the East. ... ... Zenobia, supported by her two generals, kinsmen of her husband, was now face to face with a Roman invasion. She held Egypt, Syria, Mesopotamia, and Asia Minor as far as Ancyra; and Bithynia was ready to join her party had not the army of Aurelian appeared just in time from Byzantium. She could count too on the Armenians and the Arabs, but the loyalty of Syria was doubtful: the towns disliked a rule which was essentially "barbarian," and in Antioch at least the patroness of the Monarchian bishop Paul of Samosata could to be popular with the large Christian party by whom he was bitterly hated. There were many Romans [p. 202, Col. 1c – p. 202, Col. 2a] in Zenobia's force, and it was they who bore the brunt of the two great battles at Antioch and Emesa, which followed Aurelian's rapid advance through Asia Minor. But Zenobia made light of these defeats, — "I have suffered no great loss" was her message to Aurelian, "for almost all who have fallen are Romans." ... But the empire of Palmyra came four centuries too soon. Rome was not yet exhausted, and Zenobia had neither the religious discipline of Islam to hold the Arabs together nor the spoil of the treasuries of Persia to keep their enthusiasm always fresh. Aurelian's military skill was strained to the uttermost by the prudence and energy of Zenobia, but he succeeded in forming and maintaining the siege of Palmyra in spite of its bulwark of desert, and his gold corrupted the Arab and Armenian auxiliaries. Zenobia attempted to flee and throw herself on the Persians, but she was pursued and taken, and then the Palmyrene lost heart and capitulated. Aurelian seized the wealth of the city, but spared the inhabitants, and to Zenobia he granted her life while he put her advisors to death. She figured in his splendid triumph, and by the most probable account accepted her fall with dignity, and closed her days at Tibur, where she lived with her sons the life of a Roman matron. The fall of Zenobia may be placed in the spring of 272.<ref name=":21">"W. R. S." [Smith, W. Robertson]. "Palmyra." ''The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General Literature''. Ed., Thomas Spencer Baynes. 9th edition. Vol. XVIII (Vol. 18): Orne–Phthisis. 1885. Google Books. Retrieved 11 August 2022.https://books.google.com/books?id=0au0xN1PrAEC.</ref>{{rp|p. 201, Col. 2a – p. 202, Col. 2b}} </blockquote> This extract is from "Palmyra" in the 9th edition of the ''Encyclopædia Britannica'', which does not have an article on Zenobia. The Zenobia entry in the index is a cross reference to the Palmyra article, which is about 10 columns and includes a biographical sketch of this ruler.<ref>By W.R.S., W. Robertson Smith, Lord Almoner's Professor of Arabic, University of Cambridge.</ref> <ref name=":21" />{{rp|p. 1, Col. 1c, after p. 858}} Written by George Albert Cooke<ref>Rev. George Albert Cooke (G.A.C.), Oriel Professor of the Interpretation of Holy Scripture in the University of Oxford, Fellow of Oriel College, Vol. XXVIII, p. viii. </ref>, the article on Zenobia in the 11th edition of the ''Britannica'', technically the next edition<ref>The 10th edition of the ''Britannica'', 1902–1903, is the 9th edition plus 10 supplementary volumes.</ref>, published in 1911, is substantial, more than an entire column.<ref>"G.A.C." [George Albert Cooke]. "Zenobia." Encyclopaedia Britannica: An Dictionary of Arts, Sciences, and General Information. Ed. . Vol. XXVIII (Vol. 28): Vetch to Zymotic Diseases. ''Google Books''. Retrieved 11 August 2022. https://books.google.com/books?id=NEpAAQAAMAAJ.</ref>{{rp|p. 972, Col. 1a–2b}} == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. The Duchess's entourage also included three groups, all in costume — children, trumpeters and fan-bearers. The trumpeters were "Egyptian," "negroes in full Oriental dress," as were at least some of the [[Social Victorians/1897 Fancy Dress Ball#Serving Staff|staff working at the ball]]. The people who attended the ball would have been accustomed to seeing children like black boys as attendants:<blockquote>Titian’s pictures (says a London correspondent) have made us familiar with black boys forming part of the train of certain great ladies. The Prince of Wales brought a Nubian boy home, and for a time he was in the Marlborough House and Sandringham households. The Duchess of Marlborough is very often accompanied by a black boy in Oriental scarlet and gold, who seems but to live to obey her grace’s behests; for his dark, piercing eyes follow her every movement, giving him a weird look, which is increased by the curious incision in his cheeks. At the Devonshire House fancy ball the glory of the great Oriental procession was much increased by the Nubian boys, who carried palm-leaves over some of the queenly heads.<ref>"Black Boys for London Ladies." ''Aberdeen Journal'' Wednesday 4 August 1897: 6 [of 8], Col. 5c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000032/18970804/130/0006.</ref> (“Black Boys for London Ladies”) </blockquote> === The Costumes of the Entourage === The ''Westminster Gazette'' and the ''Man of Ross'' list the groups but do not mention the number of members of the groups. According to two sources, probably in error, the London ''Daily News''<ref name=":15" />{{rp|p. 5, Col. 6a}} and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' may be the source (because it was published at 5:00 a.m.<ref name=":27">Mitchell, Charles. ''Newspaper Press Directory'', 1895. [Hathi Trust via U Wisconsin Madison.] http://hdl.handle.net/2027/mdp.39015085486150.</ref>{{rp|55}}) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century, but the report in the Belfast paper is a great deal more detailed and also came out on Saturday. These four sources describe the Duchess's retinue and how the people in it were dressed, in order by publication date and time (the first 3 coming out on Saturday 3 July 1897 and the 4th on Saturday 10 July): *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />{{rp|back right}}</blockquote> This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref name=":19">"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/visit-chatsworth/chatsworth-estate/art-archives/access-the-collection/archives-and-library/historic-servants-and-staff/.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo [[Social Victorians/People/Manchester|Montagu, 8th Duke of Manchester]] (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Douglas-Hamilton]] Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu [[Social Victorians/People/Gosford|Acheson]] (c. 1856 – 3 March 1944) #Charles William Augustus [[Social Victorians/People/Manchester|Montagu]] (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu [[Social Victorians/People/Stanley|Stanley]] (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Archives and Memoirs == * The Chatsworth Archives of the Dukes of Devonshire are at Chatsworth House. They hold the costume Louisa, Duchess of Devonshire wore as well as a few, but not many, of documents related to the ball. Perhaps most of the papers associated with the ball were destroyed, or perhaps Louisa's papers were given to one of her children? == Notes and Questions == #Louise, Duchess of Manchester was not invited to the wedding between Bertie and Alix, Victoria's punishment for Louise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref name=":28">Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #Deborah (née Mitford), 11th Duchess of Devonshire was photographed wearing the costume made for Louise, 8th Duchess in 1897 on the occasion of her 80th birthday, so perhaps 31 March 2000<ref>{{Cite news|url=https://www.thetimes.co.uk/article/wait-for-me-memoirs-of-the-youngest-mitford-sister-by-deborah-devonshire-zb3q2bk9xdq|title=Wait for Me: Memoirs of the Youngest Mitford Sister, by Deborah Devonshire|last=Athill|first=Review by Diana|access-date=2022-08-20|language=en|issn=0140-0460}} The ''Times'' 11 September 2010. https://www.thetimes.co.uk/article/wait-for-me-memoirs-of-the-youngest-mitford-sister-by-deborah-devonshire-zb3q2bk9xdq.</ref>. She wore it as Dowager Duchess of Devonshire, so she must have worn it between the time of her husband's death (3 May 2004<ref>"Andrew Robert Buxton Cavendish, 11th Duke of Devonshire." "Person Page – 959." ''The Peerage: A Genealogical Survey of the Aristocracy of Britain as Well as the Royal Families of Europe'' https://www.thepeerage.com/p959.htm#i9590 (accessed 11 August 2022).</ref>) and her own (24 September 2014<ref>"Hon. Deborah Vivien Freeman-Mitford." "Person Page — p960." ''The Peerage: A Genealogical Survey of the Aristocracy of Britain as Well as the Royal Families of Europe'' https://www.thepeerage.com/p960.htm#i9591 (accessed 11 August 2022).</ref>). #The hub for the UK National Archives bases its list of individuals on who the various archives and collections name, which means women do not always show up directly. Louise, Duchess of Devonshire does, but almost exclusively these are documents found in the Devonshire Collections, Chatsworth. #Louise, Duchess of Devonshire is #18 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] her ball in 1897. === Questions That Looking at the Actual Garment Might Answer === #Is the belt is attached to the waist of the overdress? Does it have a closure? #Is there text written into the waistband of any of the pieces? #What is the structure under the train to hold it in place? #Are the garments assembled into a single costume or are they in parts? (The train, for example, is it attached or separate?) #Were closures built in to the costume, or was it sewn onto the Duchess's body? #In the trim of the overskirt, are there really jewels in the eye of the peacock feathers, and are they really held in place by little ropes made of gold and silver cord? #The Pall Mall Gazette ['''somewhere'''] says that the "stomacher" was "formed" by a disassembled emerald tiara, but do they mean ''bodice'' instead of ''stomacher'', and ''formed'' is just bad writing with a generic verb? #The sequins appear to be tarnished silver only: no gold. Is this true? == Footnotes == {{reflist}} eduusuzkx9vi3xh8szefq3fnd0x2waj 0 264277 2835585 2835231 2026-10-02T22:59:25Z Scogdill 1331941 2835585 wikitext text/x-wiki [[Social Victorians/Timeline/1840s|1840s]] [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] 1890 [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] [[Social Victorians/Timeline/1896 | 1896]] [[Social Victorians/Timeline/1897 | 1897]] [[Social Victorians/Timeline/1898 | 1898]] [[Social Victorians/Timeline/1899 | 1899]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]] ==January 1890== Sometime in January 1890, [[Social Victorians/People/Horniman|Annie Horniman]] was initiated into the Golden Dawn (Gilbert 86 144). Sometime in 1890 Annie Besant and [[Social Victorians/People/Helena Blavatsky|Helena Blavatsky]] founded the Working Women's Club in the East End (Besant Chronology.pdf). Maye Dilke was a member (Crawford 170). [[Social Victorians/People/Frank Harris|Frank Harris]] says,<blockquote>One day in 1890 I had George Meredith, Walter Pater and [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] dining with me in Park Lane and the time of sex-awakening was discussed. Both Pater and Wilde spoke of it as a sign of puberty. Pater thought it began about thirteen or fourteen and Wilde to my amazement set it as late as sixteen. Meredith alone was inclined to put it earlier.<ref>Harris, Frank. ''My Life and Loves''.</ref></blockquote> ===1 January 1890, Wednesday, New Year's Day=== <blockquote>Sharp frost was experienced over the East of England yesterday morning, and in South London the thermometer had been as low as 27deg., and 28deg. in Hyde-park. It rose during the day to 41deg., but in the evening frost again set in. Later the thermometer rose again to 32deg., and there was an appearance as of coming rain.<ref><cite>The Morning Post. 1 January 1890 (No. 36,675): P. 4, Col. F.</cite></ref></blockquote>Influenza is spreading all over Europe. According to the "Court Circular: Orders for Court Mourning" in the Morning Post,<blockquote>A second supplement to the Gazette, issued last night, contains the following: — "Lord Chamberlain's Office, January 1. "Orders for the Court's going into mourning on Thursday next, the 2d instant, for her late Majesty the Empress of Brazil, viz." — "The ladies to wear black dresses, white gloves, black or white shoes, feathers, and fans, pearls, diamonds, or plain gold or silver ornaments. The gentlemen to wear black Court dress, with black swords and buckles. "The Court to change the mourning on Thursday, the 16th instant, viz.: — [new paragraph] The ladies to wear black dresses, with coloured ribbons, flowers, feathers, and ornaments, or grey or white dresses, with black ribbons, flowers, feathers, and ornaments. The gentlemen to continue the same mourning. "And on Thursday, the 23d instant, the Court to go out of mourning."<ref><cite>"Court Circular: Orders for Court Mourning." The Morning Post, 2 January 1890 (No. 36,676): page 5 Col. G.</cite></ref></blockquote> ===9 January 1890, Thursday=== ==== The York Hunt Ball ==== This account is from the ''Irish Society'', which explains which women's costumes are the focus:<blockquote>The fashionable world and his wife are out of town just now, having good times at country houses. They shed the light of their brilliant presence on those annual functions known as county balls, and enjoy themselves also at those popular and picturesque entertainments to which they are bidden by the masters and members of the various hunts. One of the smartest dances of last week took place at York on Thursday evening. It was in fancy costume, and was in aid of funds for the York County Hospital. Once upon a time the journey from London to the Minster City of the North was a far cry, and one not lightly undertaken. Now it is luxurious and short. In consequence this hall presented the appearance of one in town in mid-season. There was conspicuous for height and handsomeness the familiar figure and face of Countess de Grey. Lady Helen Duncombe, too, who shares with her Grace of Leinster, and Lady Cynthia Graham, the family beauty of the House of Feversham, was seen in the marvellously becoming (to her) costume of the Empire period. Lady Lilian Wemyss, a lady "divinely tall,” and appropriately fair, was garbed in pink and white of a fashion hailing from the Empire. A "square;” in which the lady dancers were all gowned empirically, was the event of the evening. An event worth remembering it was. The costumes were superb, but their attractions were nowhere in comparison with the charms the [Col. 1c–2a] wearers. These were all tall and each one handsome. Lady de Trafford’s pretty face and fine figure showed to great advantage in her gown of white satin, veiled with gold-bordered ''chiffon'', and with an Empire trained cloak from her shoulders of Rose du Barri velvet lined with white satin, and bordered with gold galon. The colour of the velvet in what is seen in the half-yellow, half-scarlet petals of the soft textured nasturtium bloom. [[Social Victorians/People/Bourke|Mrs Algernon Bourke]] looked a picture in a Gainsborough gown. The white satin skirt was flounced with sable and veiled with ''chiffon'', the setuage of which was left to show without being hemmed up. There was a broad sash of rose-pink silk and each buttonhole was filled round with crimped lisse. One might go on for long enough describing costume after costume, and then leave them half out. Therefore, we shall, like the judge "sum up” and "charge” for a verdict of the utmost brilliancy.<ref>"Our London Letter." ''Irish Society'' (Dublin) 11 January 1890, Saturday: 17 [of 24], Col. 1b–2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001939/18900111/042/0017. Same print title, p. 29.</ref></blockquote> ===16 January 1890, Thursday=== <blockquote>About two hundred members of the newly formed Dulwich Literary and Scientific Association attended the conversazione given on Jan. 16 at Surrey House Museum, Forest Hill, when Mr. Horniman, F.R.G.S., acted as cicerone round the rooms, richly stored with historical relics, ornithological curiosities, and art treasures.<ref>''Illustrated London News'' (London, England), Saturday, January 25, 1890; pg. 120; Issue 2649, Col. 3.</ref></blockquote> === 1890 January 28, Tuesday === According to the ''Dundee Evening Telegraph'' on 28 January 1890 people were buying "Exploration" shares, which funded exploration expeditions with the possibility of rights to minerals or resources or the development of monopolies for goods that were "discovered."<blockquote>Royalty and the peerage have gone in for "Exploration" shares. The Duchess of Albany is the possessor of fifty shares, which hardly seems to assume the dimensions of a royal investment. Certainly the general public would incline to the idea that a princess would never condescend to invest so small a sum as £25 in anything unless it were a hat or a jacket. The Duke of Fife, on the other hand, has been allotted a thousand shares. The Duchess of Manchester and the Countess of Mandeville have thirty each.<ref>"Notes — Mainly Personal." ''Dundee Evening Telegraph'' 28 January 1890, Tuesday: 2 [of 4], Col. 2b [of my 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18900128&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''Evening Telegraph'', n.p.</ref></blockquote> === 1890 January 29, Wednesday === [[Social Victorians/Santurce Murrieta Wedding 1890-01-29|Santurce Murrieta Wedding 1890-01-29]] === '''1890 January 30, Thursday, through at least''' 1890 February 8, Saturday === Kim, Lord Mandeville was in Brighton — at 80, King's-road — at least as early as 30 January.<ref>"Brighton Visitors." ''Brighton Gazette'' 30 January 1890, Thursday: 3 [of 8], Col. 2a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000938%2F18900130&page=3&article=073&stringtohighlight=mandeville+viscount. Print title: ''Brighton Gazette and Sussex Telegraph'', p. 3.</ref> The ''Lady's Pictorial'' lists "recent arrivals" in Brighton:<blockquote>Though a good many people have left Brighton, still fresh visitors arrive, and the front is crowded when the weather is a little kind. Amongst the most recent arrivals may be mentioned the Countess of Strafford, Lady Susan Byng, Lady Chelmsford, Lord and Lady Seaton, Lady Jane Grimston, [[Social Victorians/People/Mandeville|Lord Mandeville]], Lord Romilly and the Hon. John Romilly, Viscount and Viscountess Hampden, the Hon. R. Somerset, the Hon.  Thomas Dundas, the Hon. Sidney Greville, the Hon. A. W. Verney Cave, Lady Elizabeth Cartwright, Colonel Sir Alfred and Lady Kirby, Mr. Bazley White, M.P., and Lady Bazley White, Sir Hugh Montgomery, Sir Frederick and Lady Perkins, Lady Ellis, General Seymour, &c., &c.<ref>Dorothy. "Brighton Notes." ''Lady's Pictorial'' 8 February 1890, Saturday: 51 [of 60], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005980%2F18900208&page=51&article=671&stringtohighlight=mandeville+viscount. Same print title, p. 203.</ref></blockquote> ==February 1890== === 1890 February 6, Thursday === Before 6 February but not long before. The prize-fighter Pete Jackson fought a match at the New Pelican Club.<blockquote>It is always well to hear that distinguished visitors have enjoyed the hospitality which we have extended to them. This satisfaction is afforded us in full measure by the eminent "Parson" Davis, manager of Pete Jackson, the prize-fighter, who has just been regaling an American interviewer with an account of his experiences in England :— He had been treated in splendid style, and Jackson was received well everywhere. Nothing would give him greater pleasure than to revisit England, and he hoped soon to do so. "I shook a paw with the Prince,” said the parson. "We had a high old time with Albert E. His Royal Highness was exceptionally friendly with Jackson, and actually made a pump handle of his arm. He shook hands with him again and again. The Prince is in capital  health, and" —— But at this point the interviewer mysteriously breaks off. The Prince, however, was not the only British notable to whom Parson Davis referred. He explained that the cause of the trouble with Smith at the New Pelican Club was that the gloves furnished Jackson were too small in the fingers, and when he closed his hand he almost split them, and could not fight. The matter was explained to the officers of the Club, where the fight took place, and other gloves were procured for Jackson :— Fair play was given, and the audience, which included [[Social Victorians/People/Mandeville|Lords Mandeville]] and Lonsdale, were loud in their admiration of Jackson's skill.<ref>"The Tittle Tattle of the Week." ''Pall Mall Budget'' 6 February 1890, Thursday: 4 [of 36], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005967%2F18900206&page=4. Same print title, p. 162.</ref></blockquote> === 12 February 1890, Wednesday === ==== Lady Constance Leslie's Reception ==== <blockquote>On Wednesday last Lady Constance Leslie had an evening party at her residence in Stratford-place. The numerous guests included Prince Ernest of Hohenlohe, Princess Victor of Hohenlohe, and the Countesses Gleichen, the Austro-Hungarian Ambassador, Count Mensdorf, Count and Countess Henry Lützow, Mr. and Mrs. Henry White, Baroness de Brienen and Mdlles de Brienen, the Duke of Abercorn, the Duchess of Cleveland, the Marchioness of Blandford and Lady Frances Spencer Churchill, Earl and Countess of Portarlington, Earl Stanhope, the Earl of Northbrook, the Earl and Countess of Arran, the Earl and Countess of Erne and the Ladies Cole, Julia Countess of Jersey and Mr. Brandling, Viscount Clifden, Viscountess Bary, Viscount Torrington, Viscount and Viscountess Castlerosse and Lady Louisa Beauclerk, Viscount and Viscountess de Vesci and Lady Sarah Spencer, Viscountess Galway, and Mr Caryl Ramsden, Lord de L'Isle and Dudley, Lord Alcester, Lord and Lady Wantage, Lord Ashbourne, Lady Henry Grosvenor and Lady G. Ogilvy, Lady Cloncurry, Lord Morris and Mr. M. Morris, Lady Charles Beresford, Lord Rowton, Lady Blanche Hozier, Lady Colville, Lady Rayleigh, Lady Constance Lytton and Mr. Charles Earle, Lady Evelyn Sutton, Lady Ernest Seymour and Ladies Fortescue, Lord and Lady Sandhurst, Lady Mary Lloyd, Lady Tweedmouth, Lady Windsor, Lord and Lady William Seymour, Lady Guy Campbell, Lady Egerton, Lady Fitzgerald, Baron d'Erlanger and Mr. Charles Derenberg, Hon. R. C. Spencer, M.P., and Mrs. Spencer, Hon. Evelyn Ashley, Hon. W. Mansfield, Hon. Lionel Ashley, Hon. A. Hood, Hon. William Lowther and Miss Lowther, Hon. Sydney Greville, Hon. Fitzroy and Mrs. Stewart, Hon. Frederick Leveson Gower, Hon. Kenneth Howard, Hon. Francis Villiers, Hon. Francis Bertie, Hon. Arthur Walsh, Colonel the Hon. Charles Edgcumbe, General Hon. John Bourke, General and Hon. Mrs. Ellis and Mrs. R. Sneyd and Miss Ellis, [[Social Victorians/People/Bourke|Hon. Algernon and Mrs. Bourke]], Hon. Alan and Hon. Evan Charteris, Hon. Alexander Yorke, Hon. Derek Keppel, General Sir Dighton Probyn, Hon. Percy Wyndham, Hon. Everard Baring and Hon. Susan Baring, Hon. Mrs. Brett, Hon. Claud Hay, the Right Hon. Henry Matthews, Sir George Arthur, Sir George Maude, Sir Philip Currie, C.B., Sir Redvers and Lady Audrey Buller, Sir Charles Grant, Sir Mackenzie Wallace, Sir Herbert Maxwell, M.P., Sir Oscar Clayton, Sir Robert Gresley, Sir Algernon West and Mr. Horace West, Sir George and Lady Russell and Miss Brackenbury, Sir Henry and Lady Longley, the Sub-Dean of the Chapels Royal, General Crealock, General Charles Fraser, Colonel Saunderson, M.P., and Hon. Mrs. Saunderson, Colonel and Mrs. Stewart and Miss Streatfeild [siv], Colonel and Miss Balfour, Colonel and Mrs. Stanley Clarke and Miss Clarke, Colonel R. Baring, Captain and Mrs. Jekyll, Captain and Mrs. J. Bagot, Captain Lambert Ward, Captain A. Bagot, Mr. Charles Hall, Q.C, M.P., Mr[.] Hussey Walsh, M.P., Mr and Mrs. W. Hartpole Lecky, Mr. and Mrs. Richard Combe, Mr Henry Foley, Mrs. Arbuthnot and Miss Guthrie, Mr. and Mrs. Heseltine and Miss Dorothy Heseltine and Lord Cantelupe, the Miss Stopford Brooke, Mr. and Mrs. Thackeray Ritchie, Mr Edmund Byng, Mr. and Mrs. Wilton Phipps, Professor Sidney Colvin, Mr. and Miss Tennant, Mr. Godfrey Webb, Mr. and Mrs. Turton and Mr. Sidney Ponsonby, Mrs. Adare, Mrs Cecil Paget and Mr. Hugo Wemyss, Mrs. L. Drummond, Mr. and Mrs E. Hope, Mrs. L. Seymour and Miss Seymour, Mr. Guy Repton, Mr. and Mrs. Julian Sturgis and Miss Beresford, Mr. Poynter, R.A., Mrs. Daniel Cooper, Mr. Hamilton Aïdé, Mr. Alwyne Maude, Mrs. C. Bentinck, Mrs. Bingham Mildmay, Mr. Philip Habord, Mr. and Mrs. George Bulteel, Mrs. Beerbohm Tree and Miss Julia Neilson, Mr. Hallé [Halle?], Mr. Rudolph Lehmann and Miss Lehmann, [[Social Victorians/People/Fanny Ronalds|Mr. R. Ronalds]], Mr. Ellice, Mr. Arthur, Madame Blumenthal, Miss Julia Leslie, Mr. and Mrs. Jeune, Mr and Mrs. Charles Stuart-Wortley and Misses Millais, Mrs. Arkwright, Mr Joachim, Mr. Gillett, Mr. Pulteney (Scots Guards), Colone| Cuthbert Larking, Mr. Harry Higgins, M. Tividar Nachez, Mr. G. Ross, Herr Johannes Woolff, Mrs. Charles Bagot and Miss Bagot, Mrs. Bischoffsheim, Mr. Henry Petre, Miss Balch, Mr. and Lady Maud Ramsden, Mr. Adolphus Liddell, Mr Knowles, Mr. Grenfell, Mr. Edward Hamilton, Mrs. Henry Gordon and Miss Gordon, Mr. Henry James, Mr. St. Leger, Mrs. Dacre Hamilton and the Misses Nugent, Mr. Harbord, Mr Somerset, Mr. Ruggles-Brise, Mr. and Miss Hughes of Kinmel, Mr. and Mrs Douglas Freshfield, Mrs. Homer, Mr. W. H. Mallock, Mr. and Mrs. Robert Benson, Mr. H. Seymour, Mr. Ernest Law, Mr. Hugh Grimston, Mr. Lionel Cust, Mr. Chappell, Mr. and Mrs. Du Maurier, Mr. Boughton, R.A., and Mrs. Boughton, Mr. R. Roche, Mr. Montagu Wilson, Mr. Herbert Hope, &c.<ref>"Lady Constance Leslie's Reception." ''Morning Post'' 15 February 1890, Saturday: 2 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900215/008/0002. Same print title and p.</ref></blockquote> === 24 February 1890, Monday === Lilly Langtry starred in Shakespeare's ''As You Like It'' at the St. James's Theatre. The Daily Telegraph had a long review and mentioned more names:<blockquote> The motto of Petronius, which is said to have adorned the Globe Theatre in Shakespeare's time, "Totus mundus agit histrionem," and which is reproduced in Jaques' famous speech, may be said to have received a novel illustration in the brilliant audience which completely filled the St. James's Theatre last evening. A very distinguished section of the world "played the actor" in the stalls, boxes, and dress-circle, and evinced the keenest dramatic interest in the long-deferred production of "As You Like It." The Prince and Princess of Wales, with the Duchess of Fife, surveyed the scene from the Royal box, while amongst the many who secured seats for the first were the Austrian Ambassador, the [[Social Victorians/People/Louisa Montagu Cavendish|Duchess of Manchester]], Countess de Grey, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lady Colin Campbell, Lord Wolverton, Lord and Lady Coleridge, Sir Charles Russell, Sir Edward and Lady Guinness, Baroness de Stern, and Mr. Alfred Rothschild. Naturally enough there was much curiosity in seeing once more on the London stage an actress who has since her last appearance devoted so much serious study to her profession, and gained so much success both before American and provincial audiences; while the sympathy evoked by her recent severe illness, delaying for four weeks the production of the comedy, added an almost pathetic note to the general excitement and enthusiasm. In the second scene of the first act, when the tenderly attached cousins make their earliest appearance on the stage, Celia's first words, "| pray thee, Rosalind, sweet, my coz, be merry," answered by Rosalind’s "Dear Celia, I show more mirth than I am mistress of," were almost drowned in the applause which accompanied the recognition of Mrs. Langtry. From time to time, whenever opportunity served, enthusiasm broke out anew, culminating in the triumph of plaudits which formed the fit ending to the gracefully-delivered epilogue with which Rosalind takes her farewell of her audience. There can be no doubt that, with all due allowance for the inevitable nervousness which accompanied a first production before a critical though sympathetic house, the revival of the immortal comedy was generally pronoanced to be a success. The fancifally-named "As You Like It" — where even in nomenclature the author confesses his indebtedness to the romance of Thomas Lodge — with greater dullness than is usually the case with Shakespeare's plays. Indeed interest is hardly aroused till the scene before the Duke's Palace, where Rosalind and Celia witness the wrestling — which after a few more performances will probably go less stagily than it did last night — between Charles and Orlando. It is only when we are transported to the Forest of Arden that Shakespeare's romantic fancy allows itself full course, and it was only when she was accoutred in the doublet and hose that Mrs. Langtry exercised her full charm upon the audience. The difficulty for the actress in the character of Rosalind is the contrast between the early abandon of her frankly-expressed love for the son of Rowland de Bois and the subsequent archness and reserve with which she plays with her lover in the forest. Orlando goes through precisely an opposite change. From the careless desperateness of a man who is out of favour with fortune in the opening scene, he changes to the sentimental devotion of a "love-shaked fancy-monger" in the sylvan glades. If that interchange of parts be missed much is lost in the sprightly  contrasts of the drama. Unfortunately, no sooner did Orlando enter on the wrestling-ground last night, than he started and caught at his heart in obvious suggestion of love at first sight. Much is lost of the utter wonderment at Rosalind’s unexpected graciousness in giving him her chain, which keeps him tongue-tied later, if he betrays so early the fatal ravages of Cupid — a betrayal which is hardly likely in a man whom neglect and misery have driven to the last stage of reckless hardihood. Rosalind must love her champion before her winning thanks for his prowess have made the starred younger brother first marvel at and then adore the frank and generous advances of his benefactress. In the fine words, "My father was no traitor," Mrs. Langtry rose to the full height of the character she was depicting. Throughout the greater part of the first act there was grace, loveableness, charm, but also a kind of flatness, or, at all events, an even and somewhat uninteresting level of characterisation which did not convey the promise of all that this loveliest creation in Shakespeare's gallery could do, when the Court life was exchanged for the freer and less conventional existence in the woods. All the more praise, therefore, is Mrs. Langtry’s due, that when she finds herself in the Forest of Arden, she brings before us a truer Rosalind. The beautiful first scene between the disguised Ganymede and her amorous swain showed Mrs. Langtry at her best. The quick and nimble wit, the archness and pretty coquetry of a girl who is sick at heart with love and feeds her passion in a playful and adorable tyrannising over the unconscious lover, were depicted with a reality which won instant recognition from the spectators. For the rest, however, Mrs. Langtry’s Rosalind was an unequal performance, now rising with unexpected force into the regions of pure comedy, now sinking to an ordinary level of  uncomfortably-disguised womanishness. There were bright patches of power in the midst of a somewhat dull and colourless expanse. Much of this was, no doubt, due to the excitement and fatigue of a first night, to which also may be attributed possibly the sameness of some of her gestures and a restlessness which was something more than Rosalind's anxiety to win in her own incomparable way the happiness of which she was in search. A few nights hence firmness of carriage and self-control of manner will fill in the details of a character which was most graphically and sympathetically sketched in outline at the St. James’s last night. Mr Laurence Cautley as Orlando made most favourable impression on his audience. Through all the earlier scenes — with the exception of that wholly unnecessary lapse into amorousness on the lawn of Duke Frederick, which has been already noted — he sustained the character of a picturesque and youthful gallant with considerable success, though, possibly, a stronger dash of manly independence might not have come amiss to the Shakespearean hero. Touchstone is a more difficult part to play. Indeed, in some respects, he is the most difficult of all Shakespeare's clowns. He is on the borderland between the earlier, more conventional style of Costard, Launce, and Launcelot, and the far more elaborate and important clowns in "Twelfth Night" and "King Lear." He is not so clever as Feste, not so human and real as that inimitable and tender-hearted Edgar who tries to solace the wild ravings of his discrowned king. He is halfway between such farcical beings as Kempe and Tarlton probably depicted, and that witty, self-conscious, irresponsible Jester, which Shakespeare ultimately made an entirely new creation on Elizabethan boards. It is no discredit to Mr. Sugden that he failed to realise so ingenious a conception. He presented a Touchstone who was full of amusing quips, and he went through all the catalogue of points in the Book of Honour in the fifth act with admirable skill; the Touchstone of "As You Like It" we are probably not likely to see often on the modern stage. Mr. Bourchier’s Jaques was a far higher piece of portraiture, grave and bitter and cynical as that early disciple of Schopenhauer should be, with touches of humour which only serve to show the depths of a naturally perverse and misanthropical nature. The celebrated "Seven Ages" speech  was capitally delivered, with a conscientious care and easy mastery which are rare enough in an actor who has so lately left the ranks of the amateurs. For the rest, the Adam of Mr Everill, and the Andrey of Miss Marion Lea seemed to give most pleasure to the audience, though she was sometimes — espectally in the pronunciation of the word "poetry" — inclined to represent the boorishness of the wench with an accent  which reproduced simulated inebriation rather than native clumsiness. Miss Armbruster as Hymen also deserves a word of praise. The piece was produced under the direction of the Hon. Lewis Wingfield, and the scenery, which in the Forest Glade of Acts II., III., and IV., formed a beautiful stage picture, was designed by Messrs. Perkins and Bruce Smith. After all, was any one ever wholly satisfied with the representaton of one of Shakespeare's comedies? No thing is further from the truth as a general statement than the pedant’s theory that Shakespeare is for the study, and not for the boards. If ever, however, the paradox be true that a great dramatic artist suffers when his plays gain a dramatic representation, it is when "As You Like It," "Twelfth Night" and "Midsummer Night's Dream" are brought face to face with a modern audience and amid nineteenth century surroundings. It ts not with impunity that the light that never was on sea and land is parodied by the glare of the footlights. For, honest stage-craftsman as Shakespeare was, in the full spring-tide of his poetic genius there was a divine and masterful intolerance of the real and the commonplace, a reckless flight into an ideal world, peopled by beings coined in the mint of his dainty imagination — aerial and disembodied creations whose actions and characters refuse to submit themselves to the narrow rules of the every-day world. In such a sphere banished lovers shall be predestined to meet and cement their happy unions, and lions and snakes shall be the habitual denizens of a forest where shepherds and shepherdesses prattle their erotic strains, and even a crafty Oliver shall repent of his past sins and fall in love with gentle Celia, owing to the poetic exigencies of Cloud Cuckoo Land. How can one  see all these things with the cynical eye of the modern ''Fin de Siècle'' and not feel an inevitable shock of dissatisfaction and surprise? How is it possible to avoid the jarring sense of discord when the dream picture is realised in the setting of prosaic reality? For Shakespeare's comedy is a romantic work of the mind, a refuge for all those delightful improbabilities which an indolent or whimsical fancy strings together to construct a series of complications and incidents which interest us, while they elude the sober tests of reason and judgment. But we must not carry this vein of thought too far. It is true that we have in such plays comedies of incident rather than of character. Yet Shakespeare is too and heart-whole to linger only at the Ivory gate. He throws even into these merry plots the force of that instinctive knowledge of human nature which produces real personages, tricked out in witty and imaginative guise. Shakespeare was able to pour everything into his comedies which he did into his tragedies, though in the one there is only suggestion, while in the other there is the elaboration of that is deep and tragic as well as humorous in life. Rosalid, Orlando, Touchstone, and Jaques are none the less real because they are ideal; and it is on these grounds that we can praise the conscientious and capable artists who presented these characters in flesh and blood on the St. James's stage. Nor does Shakespeare, as some have thought, intend in this play to preach [Col. 1c–2a]  any polemic against the life of the town, or compose any passionate encomium for the simple graces of the country. He is too great to be didactic on such narrow issues. Wherever men and women live, there they can be happy; or, as Touchstone says to Corin, when he expresses ignorance of the Court, one-sided enthusiasts for either streets or hedge-rows are truly damned, "like an ill-roasted egg, all on one side." Equanimity, such that of the Duke and old Adam, carries with it, wherever it be, the gracious gentleness of an eternal Eden.<ref>"St. James's Theatre." ''Daily Telegraph & Courier'' (London) 25 February 1890, Tuesday: 5 [of 12], Col. 6a–7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001112%2F18900225&page=5&article=048&stringtohighlight=mandeville+viscount. Print title: ''The Daily Telegraph'', p. 5. </ref></blockquote> The ''St. James's Gazette'' reported:<blockquote> We came intending to like it very much, and we succeeded in our intention. Indeed, we spent a really pleasant evening at the St. James's Theatre, Many of us knew one another to nod to, and some of us had bowing acquaintance with the author of the play; so it was all very sociable and comfortable. There were just enough shop-window celebrities to give an air of distinction to the rest of the audience, and everybody had a right to hope that his neighbours were wondering who he was. The critics were quite at their ease, having nothing to criticise except the acting; not being expected to suggest any slight alterations in the play (such as cutting out Acts I., III., and V., and rewriting Acts II. and IV.), but just to accept it as it stood, without even tracing it back to the French vaudeville from which it was, no doubt, originally adapted. The performance had many individual merits, but it also had two general faults. The first and least striking fault was that the interludes were  slurred over instead of being played up to. Now, the plot of "As You Like It" is so loose that a lengthy interlude (provided it be pretty) does not check the flow of interest; indeed, it gives a welcome rest and allows you to pick up your thoughts again. The other fault is one which may be, and is likely to be, remedied the play goes on. But at present it is neither smart nor spontaneous: the parts do not hang well together, and there is none of the enthusiasm which helps to pass off the interior performances of less skilful actors. The deficiency which we have noted was probably due to the serious illness from which Mrs. Langtry had been suffering and from which it was evident last night that she had not made a complete recovery. She worked very hard as Rosalind, and worked very well, and several times she made a really good hit; she was, perhaps, at her best in her indignant appeal to the usurper Duke against his unexpected and unjust decree of banishment. She was charming, again, in her embarrassment (as Miss Litton was charming before her) when she had her first meeting with her unconscious lover and she suggested very cleverly the absence of her skirts. She did her love-making archly and merrily; but though the pretended pretence gave her a right to go beyond the limits of maiden coyness, she seemed occasionally to assume the masculine Initiative when the feminine Suggestive would have been prettier. it was a pleasant though uneven performance, with plenty of grace in it and plenty of brains. The Touchstone of Mr. Sugden was in every way admirable — crisp, humorous, and distinctive. That his clowning was occasionally tedious was no fault of his. The palate of a modern audience does not relish these long-drawn pedantries of buffoonery. Our sense of humour is not more refined — it is different, and that's the end of it. But in the scenes with Audrey (and Miss Marion Lea made a capital Audrey — open-mouthed and innocently wicked or wickedly innocent) Mr. Sugden had a better opportunity and he made the most of it. There the humour was not conventional, and the audience enjoyed a laugh which was not scholarly. A word of praise must given to Mr. Brodie for his conscientious and effective rendering of the difficult and ungrateful part of the moonstruck lover of the wayward Phoebe. The first appearance of Mr. Bourchier in London attracted much interest. As an amateur at Oxford and elsewhere he was believed Iikely to to great things upon the stage, and his performance of Jaques was by no means a disappointment. The fault of it was that he did not suggest either the melancholy or the sententiousness which we associate with the character. The melancholy was but skin-deep, and the sententiousness simply was not there. But every actor has a right to read a part in his own way if it is a good way. And there is much to said for Mr. Bourchier's interpretation of Jaques. The temptation is to be stagy; Mr. Bourchier's tendency is to be natural. And he carried it out with great success in speaking the lines of the Seven Ages of Man. He caught up the Duke's phrase just as if it had dropped in the course of an ordinary conversation; and he played with it for a time until it seemed of itself to suggest to his mind the famous soliloquy. Certainly Mr. Bourchier’s method was effective, as it drew one of the heartiest rounds of applause in the whole evening. Among the company were noticed the following well known persons:— His  Royal Highness the Prince of Wales, the Duke and Duchess of Fife, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord De Grey, Mr. Alfred De Rothschild, Lord Lurgan, Sir Charles Russell, Sir Edward Guinness, Lady Colin Campbell, Lady Curzon, Lady Londonderry, Lady Hothfield, Hon. Alan Charteris, Mr. Oscar Wilde, Mr. Akers Douglas, M.P., Mr. Justin McCarthy, M.P., Sir Allen Young, Lord and Lady Coleridge, the Austrian Ambassador, the Spanish Ambassador, Lady Hozier, Mr. George Lewis, Mr. Whistler, Lady Greville [<nowiki/>[[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]]?].<ref>"As You Like It."  ''St James's Gazette'' 25 February 1890, Tuesday: 5 [of 16], Col. 1b–c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001485%2F18900225&page=5&article=027&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> This report is from ''Blalud, The Bath Society Paper'':<blockquote>Happily, there was no disappointment in store for the friends and admirers of Mrs. Langtry, who gathered on Monday night at the St. James’s Theatre in great force, to welcome this popular lady back to London and receive her with an extra warmth of cordiality in token of satisfaction at her recovery from her recent severe illness. Eight years have elapsed since she last played Rosalind in London. Since then she has travelled much, both at home and abroad, and followed the study and practice of her art with unfailing spirit and determination. She returns to the character which all leading actresses aspire to play in Shakespeare’s glorious dream of the Court of Duke Frederick and the forest of Arden a far riper artist, and, says the ''Daily News'', she presented us on Monday with an impersonation which compared to her former efforts is as the finished portrait to the crude sketch. Her performance may lack the great high tide of girlish impulsiveness, but first nights before an exceptionally critical audience and some remains of the languor of her recent indisposition doubtless contributed to rob her acting of something of its force and impulse. But her ''Rosalind'' remains a most winning and beautiful performance. A handsomer ''Rosalind'', or one that has worn the doublet and hose with more grace and picturesqueness (Mrs, Langtry, we may here note, scorns the modern heresy of boots, and appears in the more orthodox hose cross-gartered), has certainly not been seen in our time. The performance is full of good points; as when at the words, "here on the skirts of this forest like the fringe of a petticoat," she remembers with a momentary touch of maidenly shyness that petticoats no longer form part of her attire. Again, when in bantering her lover, who exhibits so strange an inability to recognize the voice and features of the woman whom he passionately loves, she pauses at "a beard neglected," and pointing at ''Orlando’s'' smooth chin bursts into uncontrollable laughter; again, when in the mock marriage scene, with ''Celia'' for the priest, her mind for the moment seems to wander away into such a dream as this mockery might suggest to one so deeply in love. The distinguishing feature of the impersonation, indeed, is the subtle suggestion of serious passion that underlies the frolicsome and bantering moods, and exhibits itself again and again in many subtle forms. The play is beautifully put on the stage, and if some of the performers fall a long way short of ideal excellence there are several impersonations that stand forth in strong relief and are of far more than common excellence. The ''Phœbe'' and ''Silvius'' of Miss Beatrice Lamb and Mr. Matthew Brodie deserve special note: but the most brilliant success of all in the case of the minor characters was undoubtedly the ''Audrey'' of Miss Marion Lee, whose hoydenish, but not ungraceful antics, whose happy, contented ignorance, whose mingled wonderment and admiration at ''Touchstone's'' philosophical utterances were admirable. Everything that taste can suggest has been done to embellish the play; the musical features are strengthened by an efficient chorus, and the costumes designed by the Hon. Lewis Wingfield are rich and handsome. Perhaps it might be objected that Mrs. Langtry’s beautiful green velvet doublet, with its slashes, tags, and laces, her cinnamon jerkin, her black cap and jewelled buckle, and her rich and massive cloak were hardly suited to her daring enterprise on the highways and in the forests of the Duke’s dominions. The restoration of Hymen, gracefully represented by Miss Violet Armbruster, and the final morris dance in honour of the nuptials, gave the crowning fanciful touch to the performance, and prepared the spectators for the playful epilogue, which was spoken by Mrs. Langtry, in her simple nuptial white robe and leopard skin, with rare elocutionary grace and effect,. [sic] The audience which greeted Mrs. Langtry was at once distinguished and interesting. The Royal box was occupied by the Prince and Princess of Wales, the Duchess of Fife, the Princesses Victoria and Maud of Wales, and the Duke of Fife. Among the representatives of "Society" were Lady Londonderry, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord de Grey, Lord and Lady Coleridge, Lady Curzon, Lady Hothfield, Lord Lurgan, Lady Greville [<nowiki/>[[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]]?], Lady Hozier, Mr. Alfred de Rothschild, Sir Charles Russell, Sir Edward Guinness, Mr. Coningsby Disraeli, Mr. Akers Douglas, M.P., Mr. Justin M‘Carthy, M.P., Sir Allen Young, Lady Colin Campbell, Lady Monckton, Mrs. Labouchere, and the Austrian and Spanish Ambassadors. Pictorial art was represented by Mr. Whistler; music by Mdme. Antoinette Sterling; dramatic writing by Mr. H. A. Jones; and theatrical interests by Mrs. Bancroft, Miss Genevieve Ward, Mr. and Mrs. John Lart, Mrs. John Hare, Mr. M. L. Mayer, Mr. Augustus Harris, &c.<ref>"Mrs. Langtry." ''Bladud'' 26 February 1890, Wednesday: 9 [of 16], Col. 2b–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005771%2F18900226&page=9&article=055&stringtohighlight=mandeville+viscount. Print title: ''Blalud, The Bath Society Paper'', p. 9.</ref></blockquote> === 25 February 1890, Tuesday === On 5 March, the ''World'' reports on a party hosted by Mrs. Adair that must have been held the week before:<blockquote>Many smart people were present at Mrs. Adair's party in Curzon Street on Tuesday night. The rooms, lighted by electricity, looked their prettiest with floral decorations consisting entirely of tulips, the hostess appearing in a gown of terra-cotta velvet trimmed with dark fur. The Austrian Ambassador and Madame Waddington were among those who listened to Miss Damian’s rendering of a new song by Tosti, and to the glees sung by Mr. Avon Saxon's American quartet; and among others present were Lady Charles Beresford, [[Social Victorians/People/Mandeville|Lady Mandeville]], the Duchess of Marlborough, Lady Fanny Marjoribanks, who brought her sister, Lady Sarah Churchill, Sir Arthur Sullivan (just back from his annual sojourn at Monte Carlo), Lord and Lady Wharncliffe, Lady Howe and Lady Edith Curzon, Lord and Lady William Nevill, Mrs. Cavendish-Bentinck, General Charles Fraser, Lady Conyers, Madame de Brienen, and Sir Philip Currie.<ref>"What the World Says." ''The World'' 5 March 1890, Wednesday: 19 [of 44], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0006339%2F18900305&page=19&article=061&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote>The ''Daily Telegraph'' covered the party with a few additional names:<blockquote>A casual passer-by in Curzon-street last night might almost have imagined from the long string of carriages assembled there that, in spite of the cold east wind blowing, the London season had already begun. Mrs. Adair had together to hear some beautiful music, including a celebrated quartette of American singers, a large and fashionable crowd, composed of the ''élite'' of London society, comprising among many others, the Austrian Ambassador, the French Ambassadress, the Marquis and Marchioness of Tweeddale, Lady Charles Beresford, Lady Fanny Marjoribanks and Lady Sarah Spencer-Churchill, Earl and Countess of Wharncliffe, Countess Howe and Lady Edith Curzon, Lord and Lady William Nevill, Lord George Nevill, Viscount and Viscountess Curzon, Countess of Sefton and Lady Rose Molyneux, Viscountess Mandeville, Julia Countess of Jersey and Mr Brandling, Viscount and Viscountess Bury and the Hon. Mary Keppel, the Countess of Crawford, the Earl of Clonmell, the Countess of Antrim and Lady Louisa Beauclerk, Viscountess Melgund, Lady Conyers, Hon. Lady Cotterell and Miss Cotterell, Lord Alington, Lord Alcester, Lord Greenock, Viscount de Vesci, Lady Hilda Brodrick, Hon. W. Lowther, M.P., and Mrs. Lowther, Hon. Mrs. Bampfylde, Hon. Hallyburton Campbell, Mrs. and Miss Campbell, Hon. Kenneth Howard, Hon. Derek Keppel, Hon. FitzRoy and Mrs. Stewart, Lady FitzGerald, Mrs. Bischoffsheim, Madame de Brienen, Lieutenant-General Sir Seymour Blane, General C. Fraser, M.P., Mr. Chamberlain, M.P., Mr. Howard Vincent, M.P., and Mrs Vincent, Mr. Charles Hall, M.P., Mrs. Cavendish-Bentinck, Sir C. Rivers Wilson, Sir Arthur Sullivan, Sir Philp Currie.<ref>"London Day by Day." ''Daily Telegraph & Courier'' (London) 26 February 1890, Wednesday: 5 [of 8], Col. 4a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001112%2F18900226&page=5&article=094&stringtohighlight=mandeville+viscount. Print title: ''The Daily Telegraph'', p. 5. </ref></blockquote>The ''Sporting Gazette'' also had a report, on 1 March:<blockquote>Mrs. Adair's party in Curzon-street on Tuesday night was, as usual, exceedingly smart and dressy. The "smarties" turned out strong for this Anglo-American function, and wore their best frocks and jewels. Lady Tweeddale, Lady Howe, with Lady Edith Curzon, Lady Fanny Majoribanks chaperoning her younger sister, Lady Sarah Churchill, Lady Curzon, Lady Antrim with Lady Louisa Beauclerk, Lady William Nevill with some lovely diamonds, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord Alington, Sir Arthur Sullivan, Mrs Cavendish Bentinck, and others. The singing of an American quartette was ''the'' feature of the evening.<ref>"The Man About Town." "The County Gentleman." ''Sporting Gazette'' 1 March 1890, Saturday: 6 [of 38], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002525%2F18900301&page=6&article=053&stringtohighlight=mandeville+viscount. Same print title, p. 274.</ref></blockquote> ==March 1890== === 10 March 1890, Monday === The Prince and Princess of Wales hosted a ball in celebration of their wedding anniversary. This report from the ''Bedfordshire Mercury'' is very brief:<blockquote> The Prince and Princess of Wales, in celebration of the anniversary of their wedding on March 10, 1863, gave a dance at Marlborough House on Monday night. The Danish Minister and Madame de Falbe were among the company invited, also the Earl and Countess Brownlow, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lord Alwyne Compton, Baron F. de Rothschild, &c.<ref>"Notes of the Aristocracy." ''Bedfordshire Mercury'' 15 March 1890, Saturday: 7 [of 8], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001290%2F18900315&page=7&article=073&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> The report from the ''Morning Post'' suggests that this was a smaller party than usual for the Prince and Princess of Wales:<blockquote>In celebration of the anniversary of the Prince and Princess of Wales's wedding day, their Royal Highnesses gave an evening party (dancing) at Marlborough House last night. His Royal Highness the Duke of Edinburgh, her Royal Highness Princess Beatrice (Princess Henry of Battenberg) and his Royal Highness Prince Henry of Battenberg, her Royal Highness the Princess Louise of Wales (Duchess of Fife) and the Duke of Fife, his Royal Highness Prince George of Wales, her Royal Highness the Princess Victoria of Wales, her Royal Highness the Princess Maud of Wales, her Royal Highness the Princess Mary Adelaide (Duchess of Teck), and his Highness the Duke of Teck and her Serene Highness the Princess Victoria of Teck and his Serene Highness Prince Francis of Teck, and their Serene Highnesses the Prince and Princess Victor of  Hohenlohe-Langenburg and the Countess Victoria Gleichen were present. The following had the honour to receive invitations, but some amongst them were unavoidably prevented from attending: — The French Ambassador (M. Waddington) and Madame Waddington, the German Ambassador (Count Hatzfeldt) and the Countess Marie Hatzfeldt, the Turkish Ambassador (Rustem Pasha), the Austro-Hungarian Ambassador (Count Deym) and the Countess Deym, the Italian Ambassador  (Count Tornielli), the Countess Tornielli, and Mademoiselle Virginia dei Conti Lazzari, the Danish Minister (M. de Falbe) and Madame de Falbe, the Russian Chargé d'Affaires (M. Bontineff), his Serene Highness Prince Louis Esterhazy, M. A. Kroupensky, M. A. de Stoeckl, Count Leyden, Count Albert Mensdorff, Count Reventlow, Mr. and Mrs. Henry White, M. de Soveral, the Marques and Marquesa de Santurce, Mr. A. de Murrieta, Mr. C. de Murrieta. The Duke and Duchess of Leeds and Lady Alice Osborne, the Duke and Duchess of Buccleuch, the Duke and Duchess of Portland, the Duke and Duchess of Westminster, the Marchioness of Blandford and Lady Frances Spencer Churchill, the Marquis and Marchioness of Granby, the Marquis and Marchioness of Londonderry, the Marquis and Marchioness of Ormonde, the Marquis and Marchioness of Salisbury and Lady Gwendolen Cecil, the Marquis and Marchioness of Stafford, Count Gleichen, the Earl and Countess Brownlow, the Earl and Countess Cadogan and Lady Emily  Cadogan, the Earl and Countess of Clarendon, the Earl of Chesterfield, the Earl of Clonmell, the Earl of Dudley, the Earl and Countess of Dunraven and Lady Florence Wyndham Quin, the Earl and Countess Granville and Lady Victoria Leveson Gower, the Earl and Countess of Hardwicke, the Earl and Countess Howe and Lady Evelyn Curzon, the Earl and Countess of  Kilmorey, the Earl and Countess of Lathom and Lady Alice Wilbraham, the Earl and Countess of Lucan and Lady Rosalind Bingham, the Earl of Mount Edgcumbe and Lady Albertha Edgcumbe, the Earl and Countess of Romney and Lady Florence Marsham, the Earl and Countess of Rosebery, the Earl of Shaftesbury, the Earl of Stradbroke, the Earl and Countess of Sefton and Lady Gertrude Molyneux, the Earl and Countess Spencer, Viscount Drumlanrig, Lord and Lady Randolph Spencer Churchill, Lord Frederick Hamilton, Lord Ernest Hamilton, Lord and Lady Algernon Gordon Lennox, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lord Charles Montagu, Viscount Chelsea, Viscount and Viscountess Coke and the Hon. Violet White, Viscount Curzon, Lady Georgina Curzon and Lady Sarah Spencer Churchill, Viscount Fincastle, Viscount Royston, Viscount Valletort, Lord and Lady Brooke, Lord Burghersh, Lord Capell, Lord and Lady Norreys and the Hon. Coralie Glyn, Lady Charles Beresford, Lord Marcus Beresford, Lord Alwyne Compton, Lord and Lady William Nevill, Lord Richard Nevill, Lord Henry Paulet, Lord Henry Vane Tempest, Lord Herbert Vane Tempest, Admiral Lord Alcester, Lady Ampthill and the Hon. Constance Russell, Lord Alington and the Hon. Winifred and Hon. Mildred Sturt, Lord and Lady Colville of Culross  and the Hon. Blanche Colville, Lord and Lady De  Ramsey, Lady Hastings, Lord and Lady Hindlip, Lord Lurgan, Lord and Lady Revelstoke and the Hon. Susan Baring, Lord and Lady Rothschild,  Lord Rowton, Lord Suffield and the Hon. Judith and Hon. Eleanor Harbord, Lord and Lady Wimborne and the Hon. Frances Guest, Lord Wolverton, Baron Ferdinand Rothschild, Captain the Hon. Maurice Bourke, R.N., Colonel the Hon. Henry Byng and the Misses Byng, the Hon. Alan Charteris, Lieutenant-Colonel the Hon. C. Edgcumbe, the Hon. Sir Spencer and Lady Ponsonby Fane, the Hon. Sidney Greville, Colonel the Hon. Oliver Montagu, the Hon. C. R. and Mrs. Spencer, the Hon. S. M'Donnell, the Hon. Alexander Yorke, the Hon. H. and Lady Feodora Sturt, the Hon. Arthur Walsh, the Hon. R. Winn, the Right Hon. W. E. and Mrs. Gladstone and Miss Gladstone, the Right Hon. A. J. Balfour, the Right Hon. H. Chaplin, the Right. Hon. Sir Henry James, the Right Hon. Sir C. L. Wyke, the Hon. S. C. Colville, the Hon. Assheton Harbord, the Hon. Julia Stonor, the Hon. H. Stonor, the Hon. E. Stonor, Captain the Hon. Henry White, Sir Edward and Lady Guinness, Sir Charles Hartopp, Sir Frederick Johnstone, Sir John and Lady Lister Kaye, Sir James and Lady  Mackenzie, Colonel Sir Nigel and Lady Emily Kingscote, Admiral Sir R. and the Hon. Lady Macdonald and Miss Macdonald, General Sir Dighton and Lady Probyn, Sir Francis and the Hon. Lady Knollys, Major-General Sir Christopher Teesdale, Sir Oscar Clayton, Sir Allen Young, Lieutenant-General Bateson, Lieutenant-General Charles Fraser, Major-General and the Hon. Mrs. Arthur Ellis and Miss Ellis, Colonel J. P. Brabazon, Colonel and Mrs. Stanley Clarke and the Misses Clarke, Captain H. F. Stephenson, R.N., Lieutenant-Colonel F. H. Poore, Lieutenant-Colonel R. Vivian, Major C. Swaine, Captain Douglas Dawson, Captain C. H. A. Hervey, the officers of the Queen's Guard, St. James's Palace, Mr. G. Byng, Mr. Henry Calcraft, Mr. Victor Cavendish, Mr. Henry Ellis, Mr. Horace Farquhar, Mr. and Mrs. George Forbes and Miss Forbes, Mr. Montagu Guest, Mr. Charles Hall, Mr. E. W. Hamilton, Mr. and Lady Hilda Higgins, Mr. Holzmann, Miss Knollys, Mr. and Mrs. F. R. Knollys, Mr. Ronald Moncrieffe, Mr. Alfred Montgomery, Mr. Arnold Morley, Mr. Henry Petre, Mr. Alfred de Rothschild, Mr. and Mrs. Leopold de Rothschild, Mr. Reuben Sassoon and Miss Sassoon, Mr. and Mrs. Arthur Sassoon, Mr. Edgar Sebright, Mr. and Mrs. Ralph Sneyd, Mr. Christopher Sykes, and Mr. O. G. St. George Williams. Messrs. Coote and Tinney's band was in attendance.<ref>"Marlborough House." ''Morning Post'' 11 March 1890, Tuesday: 5 [of 8]. Col. 6b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000174%2F18900311&page=5&article=055&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> ==April 1890== Arnold Dolmetsch took out a three-day ticket to the Reading Room in April 1890 (Campbell 22). === 9 April 1890, Wednesday === ==== The New Forest United Hunt Ball ==== <blockquote>This annual ball was beld at the New Forest Hall, Lyndhearst, on Wednesday night, and was a most brilliant gathering, upwards of 200 being present, not only the New Forest but also neighbouring hunts being well represented. Among those present were Earl and Countess of Londesborough, Lady Mildred Denison, Lady Lilian Denison, Hon. Louis Johnstone, Hon. Gerald and Mrs Lascelles, Major Downman, Capt. Sandbach, R.H.A., Col. and Mrs Sinkins, Col. Fitzroy Smyth, Lady Thursby and party, Mr Stanley Pearce (Master of the Foxhounds), Mr F. Lovell (Master of the Deerhounds) and party, Lady Adela Goff and party, Hon. Miss St. Leger Glyn and party, Mr A. H. Jeffreys, M.P., and Mrs Jeffreys, Hon. Mrs A. Peter, Mr Somerset H. Onslow, Mr Lancelot Orde, Lord Garioch, Mr and Mrs C. K. Francis, the Countess of Normandy, Lady Mary Agar, Lord Montagu, Hon. John Scott Montagu, Viscount Cantelupe, Prince Dhuleep Singh, Hon. D. Keppel, Col. Charles Needham (1st Life Guards), [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], Mrs Cornwallis West, Mr and Mrs Shelley Bontein, Col. Gridley, Col. Newton, Col. Elliot, R.H.A., Lady Meyrick and party, Sir W. Jervois, Lady Jervois and party, &c.<ref>"The New Forest United Hunt Ball." ''The Sportsman'' 11 April 1890, Friday: 2 [of 4], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001055/18900411/024/0002. Same print title and p.</ref></blockquote> ==May 1890== Leonora Braham appeared at the Opera Comique as Ruth Ferns in ''Gretna Green''. ===21 May 1890, Wednesday=== It was his pupil the young Countess Valda Gleichen who found a sponsor for Dolmetsch's first but brief appearance at a London concert on 21 May 1890, when the 'Magpie Minstrels' … gave one of their bi-annual performances at the Princes' Hall" (Campbell 30). ==June 1890== === 3 June 1890, Tuesday === ==== Dinner and Concert Hosted by Mrs. Arthur Williams and Ball by Mrs. Menzies ==== <blockquote>Yesterday evening society passed a large proportion of time on wheels, and the number of receptions was more than the most inveterate could get through wvith comfort. ... Meanwhile, the Duke of Clarence and Avondale dined with Mrs. Arthur Wilson in Grosvenor-place, and a small and select concert followed. The Duchess of Leinster was, as ever, the most beautiful among many beauties, the Hon. Helen Henniker, after her long mourning, was warmly greeted by swarms of friends, and the Countess of Lovelace, in grey satin, chaperoned her débutante niece, Miss Milbank. Prince "Eddie" went from Mrs. Arthur Wilson's hospitable mansion to that of Mrs. J. Graham Menzies. It is unusual to find mother and daughter entertaining this illustrious young man on the same evening. Mrs. Menzies, who has succeeded Lord Suffield in his beautiful house in Upper Grosvenor-street, had one of the smartest and most successful balls of the present season. The Blue Hungarians played with immense vigour and owing to the postponement of Lady Carnarvon's ball, there was no competition for men and married women. Countess Howe, Lady Brougham and Vaux, Lady Aveland and Miss Willoughby, Viscountess Stormont, the Hon. Lady Grey Egerton and Miss Egerton, the Duchess of Buccleuch, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Earl of Craven, Countess of Coventry, Mrs. Mackay, Lady Sykes, the Marchioness of Granby, Mrs. Hwfa Williams, Lord Garioch, Sir Savile and Lady Crossley, Mr. and Mrs. Harry Lawson, and many others were present.<ref>"Yesterday Evening Society." ''Pall Mall Gazette'' 4 June 1890, Wednesday: 4 [of 8], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18900604/015/0004. Same print title and p.</ref></blockquote> ==== Münster-Hay Wedding ==== Account of the wedding of Count Alexander Münster and Lady Muriel Henrietta Constance Hay. Count Bismarck sent a gift.<blockquote>St. Andrew's Church, Wells-street, was completely filled yesterday afternoon on the occasion of the marriage of Count Alexander Münster, son of Count Münster, German Ambassador in Paris, with Lady Muriel Henrietta Constance Hay, younger daughter of the Earl and Countess of Kinnoull. The ceremony was fixed for half-past two o'clock, but all the seats in the body of the church were occupied long before that time by invited guests. The rood screen was hidden with white flowers, and a number of tall palms were skilfully arranged in and about the chancel. The bridegroom was early in his place, and wore the splendid white and gold uniform of the Garde du Corps of the Emperor; his best man, Prince Hans Heinrich Pless, wearing the scarlet and gold uniform of the Garde Hussars. The six bridesmaids were Lady Mildred Denison, Lady Dorothea Stewart Murray, Hon. Marie Hay, Hon. Marjory Murray, Miss Rosalind Lovell, and Miss Gladys Hadow (niece of the bride). They wore pretty dresses of pale blue satin, veiled with white chiffon; the skirts were edged with a frill, and the bodices finished with quaint fichus of the chiffon, fastened in front with a bunch of forget-me-nots. They wore drawn hats of the chiffon, covered with forget-me-nots, and their posies were of the same flowers, intermixed with pink roses, and tied with narrow pink ribbons. Each wore a gold bangle with centre forming the initials "A.M.," and a single turquoise drop. The bride entered the church at half-past two, and was received by her father, who led her to the chancel steps, preceded by the choristers singing a nuptial hymn. The bride's dress was of ivory satin made with a very long train. The front was fringed round the hem with orange blossoms and veiled with old Brussels lace, the gift of her mother; the bodice was entirely covered with beautiful old lace, and finished at the neck on one side with a cluster of orange blossoms. She wore a small wreath of orange blossoms and myrtle arranged in the hair at the back and a tulle veil falling almost to the end of the train and held by diamond stars, the gift of the bridegroom. The bride's other jewels included two diamond drop hairpins, the gift of Count Münster; an emerald and diamond brooch, from the bridegroom; enamel and diamond heartsease brooch, from Lady Mary Mills; a diamond and emerald bracelet, from the Duke of Beaufort; and a diamond bracelet, from the Duke and Duchess of Athole. Her train was held by her two little nephews, Masters Roland and Patrick Hadow, wearing Scotch dress with lace ruffles. The service, which was fully choral, was performed by the Rev. Provost Rorison, of St. Ninian's Cathedral, Perth, assisted by the Rev. W. T. Houldsworth, rector of St. Andrew's. The Earl and Countess of Kinnoull afterwards entertained the bridal party at the residence of the Earl of Londesborough, uncle of the bride, in Grosvenor-square. Among the numerous guests were the Russian Ambassador and Madame de Staal, the German Ambassador, the Austrian Ambassador and Countess Deym, Count Münster and the Countesses Marie, Julia, and T. Münster, Prince Pless, Prince Louis Esterhazy, Count Donhoff, Baroness von Beckendorff-Hindenburg, Baron E. Fürstenberg, Count Paul Metternich, Geheimrath Schmettau, Countess Valentin, [[Social Victorians/People/de Soveral|M. de]] [[Social Victorians/People/de Soveral|Soveral]], Captain Hasenclever, Count Leyden, Herr von Rath, Count Francis Deym, Baron and Baroness Deichmann, Count Mensdorff, Herr Pilgrim Baltazzi, Count and Countess Henry Lützow, Baroness d' Erlanger [?], Herr von Kemnitz, Baron Cramm, Vicomte de Janzé, the Duke of Beaufort, the Duchess of Athole, and the Ladies Stewart Murray, the Marchioness of Downshire, the Marchioness of Stafford and Lady Sybil St. Clair Erskine, Maria Marchioness of Ailesbury, the Countess of Dunraven and Ladies Wyndham Quin, the Countess of Kilmorey, the Countess of Galloway and Miss Stewart, the Countess of Dudley and Lady Edith Ward, the Countess of Strathmore and Lady Constance Lyon, the Earl and Countess of Roden, Lady Willoughby de Eresby and Hon. Miss Willoughby, the Countess of Westmorland. Viscount and Viscountess Raincliffe, Viscountess Coke, Viscountess Stormont and Hon. MabeI Murray, Viscountess Malden and Hon. Miss Capel, Lord Hay of Kinfauns [?], Lady Delamere and Miss Cholmondeley, Lady Brougham, Lady Esher, Lord Saye and Sele and Hon. Maud and Hon. Gertrude Fiennes, Lady Helen MacGregor and Miss MacGregor, Lord and Lady Algernon Gordon Lennox and Lady Angela St. Clair Erskine, Lady Raglan, Lord Charles Innes Ker, Lady Elizabeth Bertie, Lady Frances Pratt and Lady Clementine Pratt, Lady Geraldine Somerset, Lady Dalton Fitzgerald and Miss Payne, Lady Mary Mills and Miss Pereira, Lady Dorchester, Lady Louisa Moncreiffe and Misses Moncreiffe, Lady Norreys, Lady Petre, Lady Jane Trefusis, Lord Frederic Hamilton, Baroness de Brienen and Miss de Brienen, Helen Lady Forbes and the Misses Forbes, Lady Scott, Miss Scott and Mrs. Green, Lord Zouche and Hon. Miss Curzon, Mr. and Lady Constance Hadow, Hon. Lady Campbell, Lady William Nevill and Miss Bannerman, Hon. Alastair and Mrs. Hay, Hon. Miss Greville, Captain Hon. Randolph Stewart, Hon. FitzRoy and Mrs. Stewart, Hon. Charles Walsh, Hon. William Walsh, [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], Colonel Hon. Charles and Mrs. Hay and Miss Maude Hay, Hon. Mrs. Greville Nugent, Hon. Mrs. Colville and Miss Streatfeild [sic], Hon. Claude Hay, Sir Charles and Lady Forbes of Newe, Laely Borthwick, Sir Francis Montefiore, Sir Henry Hoare, Sir George Arthur, Sir William Gordon Cumming, General and Mrs. Wilkinson, Colonel and Mrs. Chaine, Colonel Arthur Paget, Colonel H. M. Hozier, the Provost of St. Ninian's, Major Finch, Major Candy, Colonel Lloyd, Captain and Mrs. Stewart, Mr. and Mrs. Hay Drummond, Captain and Mrs. Lloyd, Mr. William Gillett, Mr. Hulse, M.P., and Mrs. Hulse, Captain Lumley, Captain Barrow, Major Anstruther, Mrs. Hwfa Williams, Mr. Arthur Hay, Mr. Raglan Somerset, Mr. and Mrs. George Cavendish Bentinck, Mrs. Villiers, Mr. Henry Petre, Mrs. Keighley, Mrs. E. Balfour and Miss Balfour and Miss Ellice, Mrs. Dick Cunynghame and Miss A. M'Gregor, Rev. C. Weidemann, Miss Balfour and Miss E. Balfour, Mrs. and Miss Halford, Mr. and Mrs. Toler, Mrs. and Miss Townley, Rev. Dr. M'Ewan, Mr. A. F. Levita, Mr. F. Stephens, Mrs. and Miss Leigh, Mr. and Mrs. George Forbes and Miss Forbes, Mrs. Luke Wheeler, Mrs. J. Beerbohm, Dr. and Mrs. W. M. Collins, Mrs. and Miss Orde, Mrs. Murray, Miss Glyn, Mrs. Godfrey Pearse, Mr. Balfour, Mr. Leopold Goldberg, Mr. and Miss Oswa d [Oswald], Mr. and Mrs. C. Bulteel, Mrs. Brydges Wyllyams, Mr. H. Balfour, the Misses Codrington, Miss Finch, Mrs. Barron and Miss Sullivan, Mr. Hamilton Aidé, Mr. Granville and Miss Berkeley, Mr. M'Gregor, Mrs. and Miss Needham, Mr. and Mrs. A. Consett and Miss Consett, Mrs. Herbert Naylor LeyIand, Mr. and Mrs. A. Darby, Mrs. Montefiore, Mrs. Charles Murray and Miss Murray, Mrs. Thomas Legh, the Rev. John F. Mitchell, Mr. and Mrs. Claud Hay, Mr. Fiztalan Manners, Mr. Albert Stopford, Mr. Hervey Pechell, and Mr. and Mrs. Charles Orde. Later in the day Count Alexander Münster and his bride left for Hillingdon Court, Uxbridge, the residence of the Hon. A. Mills, where they will pass the early days of the honeymoon, and then proceed to Chàlet Sans Souci, Bellaggio, Lake of Como, lent by Mr. Hervey Pechell. They will afterwards go to Wiesbaden, and thence to Count Münster's German country seat, where a great reception awaits them. The wedding presents were very numerous. Count Munster presented the bride with a diamond necklace and pair of diamond drop hairpins. The bridegroom's gifts to his bride included a tiara of three diamond stars, a brooch formed of an emerald of unusual size set in diamonds, large pearl pendant, gold bracelet set with turquoises, and diamond and turquoise ring. The Duke of Beaufort presented Lady Muriel with an emerald and diamond bracelet. The Earl and Countess of Kinnoull's presents included a quantity of plate and other articles. Prince Henry of Battenberg sent a large silver-mounted claret jug; Count Bismarck, a silver box; the Marchioness of Waterford, gold snake bracelet, with pearl head and ruby eyes; the Duke and Duchess of Athole, diamond "M" bangle; Lady Ermyntrude Malet, massive silver bowl; Countess Marie Münster, silver tea set and tray; Baron E. Fürstenberg, pair of gold-mounted claret jugs; Baron and Baroness Erlanger, large silver bowl; Baron and Baroness Gungberg, pair of antique silver mounted jugs.<ref>"Marriage of Count Alexander Münster and Lady Muriel Hay." ''Morning Post'' 4 June 1890, Wednesday: 7 [of 12], Col. 7a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900604/064/0007. Same print title and p.</ref></blockquote> ===4 June 1890, Wednesday=== Derby Day. According to the Morning Post, <quote>Lady Thurlow's first dance. / Mrs. Coddington's Ball. / The Fourth of June celebration at Eton. / ... Royal Society of Literature — Paper on "Early England," by Mr. Carmichael, S. </quote> ("Arrangements for This Day." The Morning Post Wednesday, 4 June 1890: p. 7 [of 12], Col. 6A). ===13 June 1890, Friday=== ? or on the 12th? Arnold Dolmetsch had a concert at which his students performed (Campbell 31). It was reviewed very positively in the 13 June Daily Graphic, especially Helene Dolmetsch's performance. They did Purcell's Fairy Queen. They used a harpsichord. ===26 June 1890, Thursday=== There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, [[Social Victorians/People/Arthur Sullivan|Arthur Sullivan]], Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. ==July 1890== ===3 July 1890, Thursday=== <blockquote>On July 3, 1890, Besant’s house at 19 Avenue Road, St. John’s Wood, was inaugurated as a new center of Theosophical work, serving as the European headquarters of the [[Social Victorians/Theosophical Society|Theosophical Society]] and as HPB’s residence. That house became also the meeting place of HPB’s Inner Group, twelve of her students who were eager for more intense training than was feasible in a larger, more general gathering.</blockquote>That Inner Group is different from the Esoteric Section.<ref>"Esoteric World Chapter 23." ''The Theosophical Society of America, Online Resources''. http://www.theosophical.org/index.php?option=com_content&view=article&catid=23&id=1725&Itemid=53</ref><blockquote>The hall had a seating capacity of about 200. On the opening night [July 3, 1890] [sic] the room was crammed, and many were unable to gain admission. The speakers were Mrs. Besant, Mr. Sinnett, a Mrs. Woolff (of America), and Mr. [Bertram] Keightley. HPB was present but said nothing, on account of the critical state of her health.<ref>"23a. Henry S. Olcott, July 1890, London [Olcott 1931, 4:254–6]." Esoteric World Chapter 23. ''The Theosophical Society of America, Online Resources''. http://www.theosophical.org/index.php?option=com_content&view=article&catid=23&id=1725&Itemid=53.</ref></blockquote> === 4 July 1890, Friday, 11 p.m. === ==== The Queen's State Ball at Buckingham Palace ==== The Prince and Princess of Wales were hosts of this ball, as Queen Victoria was not present. Many of the Queen's children and their spouses were present.<ref>"State Ball at Buckingham Palace." ''Morning Post'' 5 July 1890, Saturday: 5 [of 8], Col. 5a–7b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900705/038/0005. Same print title and p.</ref> The dancing commenced shortly after the Prince and Princess of Wales arrived at 11:00. === 1890 July 15, Tuesday === ==== Garden Party at Marlborough House to Meet the Queen ==== The beginning of the ''Morning Post'' article, which is followed by a very long list of people invited:<blockquote>Their Royal Highnesses the Prince and Princess of Wales gave a garden party at Marlborough House yesterday afternoon to meet the Queen. Her Majesty was accompanied by their Royal Highnesses the Princesses Victoria and Margaret of Prussia, and was attended by the Duchess of Roxburghe, Miss M'Neill, the Hon. Evelyn Moore, the Hon. Bertha Lambert, and Major-General Sir Henry Ewart and Major F. A. Bigge. Madlle. De Perpigna was in attendance on the Princesses Victoria and Margaret. The following were present: — Their Royal Highnesses the Duke and Duchess of Edinburgh and Prince Alfred, and the Princesses Marie, Victoria, Alexandra, and Beatrice of Edinburgh; their Royal Highnesses the Duke and Duchess of Connaught and Prince Arthur, and the Princesses Margaret and Patricia of Connaught; their Royal Highnesses the Prince and Princess Christian of Schleswig-Holstein Sonderburg-Augustenburg and Princess Victoria and Princess Louisa of Schleswig-Holstein, her Royal Highness the Princess Louise, Marchioness of Lorne, and the Marquis of Lorne. Her Royal Highness the Duchess of Albany, her Royal Highness the Princess Frederica and the Baron von Pawell-Rammingen, his Royal Highness the Duke of Cambridge, her Royal Highness Princess Mary Adelaide Duchess of Teck and his Highness the Duke of Teck and Princess Victoria of Teck, his Royal Highness the Duke d'Orleans and Princess Hélène d'Orleans, his Imperial Highness Don Pedro Augustus of Coburg and Braganza, his Serene Highness the Prince of Schaumburg-Lippe, his Serene Highness the Prince of Leiningen and their Serene Highnesses Prince and Princess Victor of Hohenlohe-Langenburg and the Countesses Gleichen, and Count Gleichen.<ref>"Garden Party at Marlborough House." ''Morning Post'' 15 July 1890, Tuesday: 5 [of 8], Col. 3a–7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900715/044/0005. Same print title and p.</ref></blockquote> === 19 July 1890, Saturday === ==== Wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner ==== This report from the ''Fife Free Press'' is missing a big piece of the 2nd paragraph, where the bridesmaids' and bride's dresses are described, because the page was damaged. The digitized copy is also generally poor.<blockquote>On Saturday forenoon, in Michael’s Church, Chester Square, London, was celebrated the marriage of Lord Loughhorough, eldest son of the Earl of Rosslyn with Miss Violet Aline Vyner, younger daughter of Mr R. C. Vyner, of Fairfield, Yorkshire. The chancel was beautifully decorated with palms and a profusion of lilies most artistically arranged. The Prince of Wales, Duke of Cambridge. Princess Mary Adelaide, Duchess and the Duke of Teck, and Princess Victoria, and the Right Hon. W. E. Gladstone were among those present. The wedding party began to arrive at 11 o’clock, the bridesmaids being early in attendance. They were the Ladies Sybil and Angela St Clair Erskine, sisters of the bridegroom; Lady Anne Lambton, Miss Musgrave, Miss Rachel Gurney, and Miss Fleming. They were charmingly dressed in soft cream crepon, with violet velvet Zoave jackets, forming half sleeves and capes at the back, and pretty turn-down collars and cravats of white mosseline [mousseline?] chiffon, and the skirts were very plainly made. They wore large white hats lined with a band of violet velvet near the face, and trimmed on the crown with white ostrich plumes, a bunch of violets fastening the brim at the back. Each wore a brooch — the St Clair cross in rubies and diamonds set in a circle of the latter, and surmounted by a baron’s cornet of the same precious stones — the gift of the bri[degroom]; and carried a bouquet of Marechal Niel [roses?] ...ging sprays of the same, and tied w... ribbons. The bride entered the ... eleven o’clock, being received at ... father, who led her to the cha... her away. The service be... procession was preceded up ... choir, who sang "The voic..." The bride’s dress was ... with a long court train, ... white velvet being embro... ...und the train, and fasten... ...ver true lover’s knot. The picture... ...gh silver waistband, had full slee... mousseline chiffon; and a small ... blossoms was most becomingly arra... with a long tulle veil. The Earl of Chesterfield discharged the duties of best man. The Rev. Canon Fleming, Chaplain in Ordinary to the Queen and Vicar of St Michael's, performed the marriage rite, assisted by the Rev. Edgar Sheppard, Sub-Dean of the Chapels Royal, and the Rev. J. Thomson, of Rosslyn, near Edinburgh. The hymn “Thine for ever! For of love,” was sung the end of the service. Canon Fleming afterwards delivered a most impressive address. After the benediction the clergy and bride and bridegroom passed to the vestry, the Prince of Wales following with Mrs Vyner. Pretty wedding favours of orange blossoms, Parma violets, and white heather were then distributed amongst the congregation. Mr and Mrs Vyner afterwards welcomed the wedding party at Ashburnham House, Dover-street. The ''dejeuner'' was served in three rooms, one being reserved for the bride and bridegroom and the Royal party. The table in the latter room was decorated with several of Mr Vynet's racing trophies. The band of the Grenadier Guards performed in the garden during the reception. The Prince of Wales, Duke of Cambridge, the Duke and Duchess of Teck and Princess Victoria were present at the breakfast, and among the general company were Count Munster and Countess Maria Munster, the Marchese and Marchesa de Montagliari, the Marquis and Marchioness of Ripon, the Marquis and Marchioness of Stafford, the Marquis of Cholmondeley, the Marquis of Northampton, Maria Marchioness of Ailesbury, the Dowager Marchioness of Conyngham and Lady Blanche Conyngham, the Marchioness of Downshire, the Countess of Rosslyn, Earl and Countess Compton, the Earl of Feversham, the Earl of Durham, the Countess of Coventry and the Ladies Coventry, Countess Howe and Ladies Curzon, the Countess of Yarborough, the Earl of Clonmell, the Earl and Countess of Loudoun, the Countess of Kilmorey, Countess de Grey, Viscount Molyneux, the Bishop of Gibraltar and Mrs Sandford, Viscountess de Labrosse and Mdles. de Labrosse, Lord Greenock, Lord and Lady Algernon Lennox, Lady Lilian Wemyss, Lady Julia Wombwell and Miss Wombwell, Hon. Lady Campbell of Blythswood, Lady Constance Harris, Lady Florence Harris, Lord and Lady Burghley, Lady Dorchester, Lord Houghton, Lady Curzon, Lady Brougham and Vaux and Miss Musgrave, Baron de Hirsch, Helen Lady Forbes and the Misses Forbes, Baroness Burdett Coutts and Mr Burdett Coutts, M.P., Lady Astley, Lord and Lady William Nevill, Lord and Lady Brooke and Hon. Leopold and Hon. Marjorie Greville, Lady Dalton Fitzgerald and Miss Payne, Lord Elcho, Lady Willoughby de Eresby and the Misses Willoughby, Lady Ewart, Lady Hindlip, Lady Cecilia Rose, Lord Rowton, Lord Capell, Lady Mabel Compton, Lord and Lady Herries, Lady Sykes, Lord and Lady Alwyne Compton, Lady Ventry and Hon. Miss de Moleyns, Hon. Lady Knollys, Lady Manners, Hon. Lady Drummond and Misses Drummond, Hon. Sydney Greville, Hon. Assheton Harbord, Hon. Derek Keppel, Hon. Captain Bourke, Hon. Mrs Percy Mitford, Hon. Philip Harbord, Hon. and Rev. L. Tyrwhitt, Hon. Coralie Glyn, Hon. M. Lambton, Hon L. Greville, Hon. Alwyn Greville, Hon[.] Eva Greville, Hon. Frances Dawnay, Hon Evelyn Paget, Hon Mrs Algernon Grosvcnor, Hon and Mrs Lambton. Hon. Mrs Evelyn Pelham, Hon. Fitzroy St Clair Erskine, [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], Hon. W. Maxwell, Hon. G. and Mrs Dawnay, Hon. Captain Lambton, General Sir Seymour Blane, Sir William Gordon Cumming, Sir Archibald and Miss Edmunstone, Sir Henry and Lady Meysey Thompson, Mrs Gladstone, Mr and Lady Margaret Cameron, Mr and Lady Florence Astley, Mr and Lady Elizabeth Melville Cartwright and Miss Melville Cartwright, General and Mrs Duff and Miss Duff, General Owen Williams, Colonel and Mrs Anstruther Thompson, Colonel Stopford Sackville, Colonel Craig, Colonel Brabazon, Colonel Forester, Colonel Stanley Clarke (in attendance on the Prince of Wales) and Miss Clarke, Major Davidson, Canon and Mrs Fleming, Captain St John Mildmay (in attendance on the Duke of Cambridge), Captain and Mrs FitzGeorge, Mrs William Portal, the Sub Dean of the Chapels Royal, Rev. W. M. and Mrs Wollaston, Rev. R. West, Mr and Miss Chamberlain. Mr and Mrs W. H. Grenfell, Mr Christopher Sykes, M.P., Mr Reuben Sassoon, Mr Arthur Sassoon, Mrs Bischoffsheim, Mr and Mrs Herbert Naylor Leyland, Dr and Mrs Anderson, Mrs Oswald, Mr and Mrs Surtees, Mr Fitzgerald, Mr and Mrs Menzies, Miss Willoughby and Miss Gertrude Willoughby, Mr Arthur Paget, Miss Christie, Mr Dowdall, Mrs Eastwood and the Misses Eastwood, Mr Frederick Warre, Mr William Williamson, Mrs Burton, Mrs Chaine, Captain and Mrs Livesey Wardle, Captain A. R. Hill, Miss Helby, Mrs and Miss Winslow, Mr and Miss Oswald, Mr Willoughby, Mr Quintin Hume, Mr F and Miss Fitzherbert, Mrs and Miss F. Hotham, Mr and Mrs Moreton Frewin, Mrs Charles Otway, Mr Calcraft, Mr Watson, Mr Fennick, Mr Montgomery, Mr Cecil Murray, Mr and Mrs Sneyd, Mr Leslie Stephen, Mr Edgar Sebright, Miss Fleetwood Wilson, Mrs and Miss Farmer of Nonsuch, Mr and Mrs Dennistoun and Miss Dennistoun, Mr and Mrs J. R. Lane Fox and Miss Lane Fox, &c. The bride and bridegroom left early in the afternoon for Easton Lodge, Dunmow, the seat of Lord and Lady Brooke, in Essex, where they will spend the honeymoon. The bride travelled in a costume of biscuit coloured armure silk, the top of the bodice and sleeves being of pale pink mousseline de soie, and the lower part of the former inserted with pink velvet ribbons to form a high band, Tuscan straw-hat trimmed with pink crepe de chine, and biscuit feathers. The presents exceeded four hundred in number. The Prince of Wales presented the bride with a trefoil moonstone and diamond brooch and the bridegroom with a sapphire and diamond pin. The Princess of Wales's gift the bridegroom was a silver-necked claret jug. The Duke of Cambridge gave an amethyst and diamond pendant, and the bridegroom a pair of silver gilt candlesticks; and Princess Mary and the Duke of Teck presented the bride with a silver sugar basin. The bridegroom’s gifts to the bride were a diamond and sapphire watch bracelet, a diamond half-hoop ring, diamond brooch with pearl centre, diamond and turquoise brooch, formed of the St Clair badge in rubies and diamonds set in a circle of diamonds, surmounted by a baron’s coronet in the same precious stones, a large salts bottle with two heart-shaped moonstones set with rubies and diamonds and a silver sugar basin. The bride presented Lord Loughborough with a turquoise and diamond stud. The Earl of Rosslyn presented the bride with diamond necklet. The Countess of Rosslyn’s gifts were diamond orchid brooch and a pink feather fan. Mr R. C. Vyner’s gifts to his daughter were a diamond and pearl tiara and gold mounted dressing case, and Mrs Vyner gave her a sapphire and diamond ring. The Marchioness of Stafford gave a ruby and diamond pin; Lady Mary Vyner antique diamond tiara; the Ladies Angela and Sybil Erskine sapphire and diamond bracelet; Hon. Fitzroy St Clair Erskine with "1890" in rubies and diamonds; Lord and Lady A. Compton large diamond star; the Earl of Rosslyn presented his son with a large silver bowl and two silver dishes; the Countess of Rosslyn's present was a travelling bag; Mr Vyner gave him a luncheon basket; Mrs Vyner a silver ''entree'' dish; the Marchioness of Stafford case containing silver coffee and tea pots, urn and stand, sugar basin, and cream jug; Lord and Lady Brooke four antique silver sauce boats; Lord and Lady Algernon Gordon ['''Col. 1c–2a'''] Lennox three antique silver ''entree'' dishes. The Earl of Rosslyn's Fifeshire tenants presented Lord Loughborough with a beautiful antique silver bowl, two antique silver dessert dishes, and two antique sugar castors; and to the bride they gave an antique silver tea service. The congregation of Rosslyn Chapel presented the bridegroom with a complete and elegant writing set, whilst Lord Rosslyn's servants gave him a silver inkstand and pair of silver candlesticks. Mr Vyner's servants presented the bride with a silver salver and six ''liqueur'' cups. The servants of Newby Hall gave a silver coffee pot, and Mr Vyner’s Cheshire tenants presented a case containing silver tea pot, sugar basin, and cream jug. The bride also received from the Duchess of Portland a travelling bag; the Duchess de Luynes, sapphire and diamond ring; the Duke of Westminister, Indian necklace; the Marquis of Northampton, diamond bracelet; the Marquis of Ripon, sapphire and diamond bracelet; the Marchioness of Ripon, turquoise and diamond bangle, the Earl of Durham, a diamond half hoop bracelet; Earl and Countess Cowper, a diamond "V V” brooch; Dowager Viscountess Downe, moonstone and diamond crescent; Lady Skyes [Sykes] ruby and diamond fish brooch; Lady Millbank, sapphire and diamond brooch Lord Douglas Compton, double heart brooch; Lord Herbert Vane Tempest, gold curb chain bracelet set with pearls; Earl and Countess Compton, bangle with “Vi” and “good luck” on it; the Duke and Duchess of St Albans, tea service with table; the Marchioness Conyngham, silver cream jug; the Marchioness of Tavistock, silver vase; the Countess of Rodin, a silver I bowl; the Countess of Yarborough, silver candlesticks; Viscountess Downe, silver tea caddie; Lady de Trafford, gold mounted salts bottle; the Hon. F. Dawnay, large silver candlesticks[;] Hon. F and Lady Lampton, travelling clock; Hon. W. Lambton, silver candlesticks; Sir R. and Lady Graham, silver salver; Sir Roderick and Miss Cameron, two silver dishes; General Williams, diamond bangle; Colonel and Mrs G. FitzGeorge, gold mounted bottle; Major and Mrs Egerton, pair of silver candlesticks; Major Stuart Wortley, diamond Swallow; Major Davidson, heart-shaped brooch in coral and diamonds; Captain and Lady Cecilia Rose, a brougham; Mr Beaumont, silver bowl; Miss Edmonstone, moonstone and diamond hat pins; Mrs Harris, sapphire and diamond bracelet; Mr Egerton, turquoise and diamond brooch; Sir Cecil Murray, sapphire and diamond pin; Mr A. Farquharson, diamond swallow brooch; Mr and Mrs Hulse, two small silver vases; Miss Fleetwood Wilson, diamond bangle; Mrs A. Wilson, tortoiseshell candlestick; Mr Quintin Hume, salts bottle mounted with diamonds and rubies; Miss Helbert, gold violet pin; Mr Chamberlain, pair of silver candlesticks; Hon. Arthur Coventry, gold mounted bottle with "Vi" in diamonds; Mrs Leopold de Rothschild, diamond hand brooch, together with a number of other gifts. Among the presents to the bridegroom were — from the Marchioness of Downshire, antique silver sugar sifter; the Marquis of Exeter, silver sandwich case; the Marchioness of Bath, antique Dutch silver box; the Marquis of Downshire, a gold pencil; Count Munster, cigarette case and match box; Count Alexander Munster, a hunting flask; the Countess of Lindsay, antique silver pen-tray; the Earl of Mar and Kellie, silver matchbox; the Earl and Countess of Loudoun, silver cigarette box; the Earl of Chesterfield, antique silver gilt inkstand; the Countess of Bradford, gold seal with agate; the Countess of Warwick, antique silver bowl; the Earl of Coventry, silver pen-tray; the Countess of Wharncliffe, two silver candlesticks; the Earl and Countess of Cromartie, two antique silver cups; Viscountess of Newport, silver paper knife; Mr and Mrs A. Sassoon, ruby and diamond pin; Lord John Cecil, silver salver; Lord and Lady Willoughby de Eresby, antique silver-mounted cruet stand; Lord William Nevtll, antique brandy carrier; Lady Margaret Spicer, silver match box; Lord Burghley, four silver salt cellars and spoons; Baron de Hirch [Hirsch?], silver inkstand; Lady W. E. Osborne Elphinstone, silver inkstand; Lord Kesteven, silver cream jug; Lord Capell, silver cigarette case; Lord and Lady Gerard, tortoiseshell and silver travelling clock; Lady Lilian Wemyss, silver cigarette case; Dad Alwyne Compton, umbrella containing matchbox and whistle; Lord and Lady Lennox, two antique silver ''entree'' dishes and bacon dish; Lord Burghersh, silver cigarette lighter; Lady Angela 8t Clair Erskine, four silver salt cellars; Lady Randolph Churchill, gold pencil case, with two sapphires; Lord Houghton, antique gold basket; Lady Sybil St Clair Erskine, umbrella; Lord and Lady Dorchester, large crystal seal; Hon. Derek P. Keppel, pair of silver candlesticks; Hon. E. and Mrs Pelham, silver inkstand; Hon. G. N. Curzon, silver gilt bowl; Hon. F. St Clair Erskine, four silver candle-sticks; [[Social Victorians/People/Bourke|Hon. A. and Mrs Bourke]], four small silver dessert dishes; Hon. Mrs Elliot Yorke, pair of silver candlesticks; Hon. Mr and Mrs L. Greville, tortoise-shell and silver box; Hon. R Somerset, diamond and pearl pin; Hon. G. H. Willoughby, silver ink-stand; Hon. H. and Mrs Grosvenor, umbrella; [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], enamel and diamond pin; Captain Hon. Maurice Bourke, cigarette case and lighter; Hon. W. Maxwell, silver-backed clothes and hat brushes; Hon. Sidney Greville, saltspoons; Hon. M. C. Maxwell Scott, two silver ''menu'' plates; Hon. W. Edwardes, pin; [[Social Victorians/People/Greville|Captain Hon. A.]] and [[Social Victorians/People/Lady Violet Greville|Mrs Greville]], silver gilt bowl, fork, spoon, and ladle; Sir Frederick Milner, diamond and moonstone pin; Sir George Chetwynd, two long silver flagons; Sir M. and Lady Fitzgerald, two silver saltspoons; Sir N. Gordon Gumming, silver-handled paper knife, &c.<ref>"Marriage of Lord Loughborough with Miss Vyner." ''Fife Free Press'' 26 July 1890, Saturday: 2 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001110/18900726/015/0002. Same print title and p.</ref> </blockquote> === 24 July 1890, Thursday === ==== Dinner and Dance Hosted by Lord Alington ==== Perhaps the date on this is uncertain, since the article in ''Truth'' would suggest it was held in late May or early June. The newspaper story in the ''Queen'' mentions another occasion in which the flowers were notable as well. From ''Truth'':<blockquote> But not a quarter of a mile distant [from a dinner and dance hosted by Lord Rothschild featuring Princess Louise and the Marquis of Lorne] a more important event was proceeding, for Lord Alington, having previously walked down the Row and invited his friends right and left to Alington House, South Audley-street, proceeded to give a dance which was honoured by the presence of the Prince and Princess of Wales, the Princesses Victoria and Maud, the Duke of Clarence, Princess Mary Adelaide, and Princess "May" and Prince and Princess Christian. This without exaggeration, with the exception of Lady Leconfield's, Lady Willoughby de Eresby's, and Lady Cadogan's, was the best ball hitherto given this season, and the prolific way ortolans [small songbirds regarded as a delicacy] were distributed at supper will long linger in the memories of those by whom these delicacies are cherished. One lady displayed her enthusiastic appreciation by devouring five of these, and it is to be hoped her devotion did not endanger her life. As for beauty, seldom before have so many pre-eminent for good looks been present at one gathering, as a mere enumeration of a few of these will prove. Of course the Duchess of Leinster heads the list, followed in quick succession by the Duchess of Portland, Lady de Grey, Lady Randolph Churchill, Lady Dudley, Lady Kilmorey, Lady Brooke, Lady de Trafford, Lady Helen Duncombe, Lady Hilda Dundas, Lady Sybil St. Clair Erskine, Lady Olivia Taylour, Miss Rachael Gurney, Lady Cardross, Miss Bourke, Lady Rose Leigh and her sister-in-law, Miss Leigh, Miss Berens (sister to Lady Cairns), Mrs. Ralph Sneyd, Lady Gerard, and a host of others far too numerous to mention. I often wonder that no painter has yet attempted to give us in one picture a permanent record of the chief beauties of our day. It would make a most interesting and exquisite group of lovely women, such as could not be found probably in any other country than this.<ref>Marmaduke, pseud. "Letters from the Linkman." ''Truth'' 12 June 1890, Thursday: 26 [of 56], Col. 1a–b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18900612/064/0026. Same print title, p. 1212.</ref></blockquote> From the ''Queen'':<blockquote>The P<small>RINCE AND</small> P<small>RINCESS OF</small> W<small>ALES</small> and the Princesses Victoria and Maud honoured L<small>ORD</small> A<small>LINGTON</small> by their presence at D<small>INNER</small>, at A<small>LINGTON</small> H<small>OUSE</small>, South Audley-street, on the 24th ult., when a distinguished party was present to meet them, including, among others, the Duke and Duchess of Leinster, the Marquis of Hartington, the Marquis and Marchioness of Ormonde, Maria Marchioness of Ailesbury, Earl and Countess Cadogan and Lady Emily Cadogan, the Earl Countess of Lucan and Lady Rosaline Bingham, the Earl of Hardwicke, the Countess of Dudley and Lady Edith Ward, the EarI of Chesterfield, the Earl of Stradbroke and Lady Hilda Rous, and Earl Annesley. Covers were laid for sixty. Lord Alington afterwards gave a dance to meet the Prince and Princess of Wales and their daughters. As on the previous occasion, the floral decorations were exquisite. The entrance hall and the approaches to the reception rooms were decorated with tall palms and a profusion of lilies, pelargoniums, ferns in every variety, and roses in every imaginable shade. The conservatories at the back were carpeted and furnished luxuriously, and blocks of ice, lighted by fairy lamps, were placed about with artistic effect, serving to keep the atmosphere cool and fresh. Feathery palms, hydrangeos [hydrangeas] and arums were here used freely, and a particularly pretty result was obtained by the windows being framed, one in a fringe of ferns and red pelargoniums, and the other in ferns and fuschias. The ballroom fireplaces were adorned on each side with palms reaching to the ceiling, the mantelpieces being hidden by a delicate arrangement of roses in various shades springing from banks of hydrangeas, calceolarias, pelargoniums, and marguerites, and the walls of the staircase leading from the conservatory to the ball-room were hung with half-balls of floral roses. The supper was served in the dining-room, the tables being embellished with the loveliest orchids and roses of the most delicate tints, intermixed with pancretia; the arrangement was so light and elegant that it was difficult to believe that hands had ever touched the flowers. On those tables with silver candelabra in the centre, the flowers were all pure white, and those having gold candelabra were treated with mauve orchids and the palest pink and yellow roses. A tent, arranged near the conservatory for the Prince of Wales, was draped with Oriental hangings, and outside were groups of flowers and myriads of fairy lamps. The company began to arrive at eleven o'clock, when dancing commenced to the strains of the Blue Hungarian Band. The Princess of Wales was dressed in white striped satin and silk, the front being embroidered in pale shades, and wore handsome diamond ornaments. The Princesses Victoria and Maud wore white tulle over satin, the front being sprinkled all over with tiny sprays of lilies of the valley, each laid on a single leaf. The Duchess of Wellington was in black satin, with a pink sash arranged very prettily across the back of the bodice, and falling to the feet; a cluster of pink flowers and a beautiful diamond spray were fastened in the front of the bodicea and a diamond rivière was arranged in the hair, a diamond necklace being also worn. The Duchess of Westminster was in white satin trimmed with gold embroidery, and veiled in white chiffon. The Marchioness of Ormonde wore white striped brocade and satin trimmed with white lace, and a splendid diamond tiara and other ornaments. Countess de Grey, palest blue satin, with front beautifully embroidered in beads, and a diamond and pearl coronet. Maria, Marchioness of Ailesbury, white brocade and lace, with splendid diamond and emerald ornaments. The Marchioness of Bath, cream an violet brocaded velvet, the front being covered with mauve [38, Col. 3c – 39, Col. 1a] violets. Lady Mary Mills, rose-pink satin, with front richly embroidered; Lady Cardross, white satin, trimmed round the bodice and train with silver embroidery; Lady William Nevill, pale yellow brocade, the front veiled with black net, trimmed up the side with jet leaves, and all round the skirt was a bordering of Maréchal Niel roses, over a flounce of black net; Countess Granville wore rich white brocade and beautiful jewels; the Countess of Sefton, black satin and diamonds; Viscountess Dungan, white mousseline de soie, the skirt edged with a deep flounce; Lady Brooke, pale blue satin, trimmed with silver; Lady Emma Crichton, white satin train, with front of white lisse, prettily embroidered in red; Lady Idina Brassey, pale mauve bengaline, with front of chiffon and a handsome diamon coronet; Lady Rose Molyneux, white cross crêpe de Chine, with a pale green sash, and prettily trimmed with pink roses; Lady Wimborne, pale grey satin; the Hon. Miss Guest, pink satin, veiled with white net, sparkling with silver; Lady Sarah Spencer Churchill, white satin, with a chiffon frill; Lady Chesham, white silk, with pink roses; the Countess of Normanton, pale grey satin, with front of fine old lace; Mrs Hulse, handsome white brocade, diamond tiara, &c.; Mrs Bischoffsheim, white silk, embroidered with silver, and trimmed round the bottom of the skirt and train with a ruche of pale yellow crèpe; the Hon. Muriel Brassey, white chiffon, over satin, slightly trimmed with pink; and the [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]] was in pink, with magnificent diamonds.<ref>"Entertainments, Balls, &c." ''The Queen'' 2 August 1890, Saturday: 38, Col. 3c – 39 [of 82], Col. 1a–1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18900802/237/0038 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18900802/237/0039. Print: ''The Queen, The Lady's Newspaper'', pp. 170–171.</ref></blockquote>This is the story from the ''Morning Post'':<blockquote>The Prince and Princess of Wales and the Princesses Victoria and Maud honoured Lord Alington by their company at dinner last evening at Alington House, South Audley-street. The guests present to meet their Royal Highnesses comprised the Duke and Duchess of Leinster, the Marquis of Hartington, the Marquis and Marchioness of Ormonde, Maria Marchioness of Ailesbury, Earl and Countess Cadogan and Lady Emily Cadogan, the Earl and Countess of Lucan and Lady Rosaline Bingham, the Earl of Hardwicke, the Countess of Dudley and Lady Edith Ward, the Earl of Chesterfield, Earl Annesley, the Earl of Stradbroke and Lady Hilda Rous, Lord and Lady Gerard, Lady Norreys, Lord Rowton, Lord Calthorpe, Lord Frederick Hamilton, Lord Charles Montagu, Lord Herbert Vane Tempest, Lord Arthur Grosvenor, Lady Lilian Wemyss, Colonel the Hon. Oliver Montagu, Sir John and Lady Lister Kaye, the Hon. Humphrey and Lady Feodore Sturt, the Hon. John Baring, the Hon. Henry and the Hon. Julia Stonor, the Hon. Gilbert Willoughby, Lady Suffield and Major-General Ellis, in waiting on the Prince and Princess of Wales; Sir Henry Calcraft, Mr. and Lady Rose Leigh, General and Mrs. Owen Williams, Mr. Mildmay, M.P., Mr. and Mrs. Arthur Sassoon, Mr. Montague Guest, Captain Orr-Ewing, Mrs. Bulkeley, Mr. Ricardo, and Mr. Caryl Craven. Lord Alington afterwards gave a dance expressly to meet the Prince and Princess of Wales. Among the company were— The Russian Ambassador, the Danish Minister, Prince Louis Esterhazy, Comte Costa de Beaunyardt, the Duchess of Buccleuch and Lady Evelyn Fitzmaurice, the Duchess of Westminster, the Dowager Duchess of Marlborough and Lady Sarah Spencer Churchill, the Duchess of Wellington and Lady Cardross, the Marquis and Marchioness of Granby and Miss Bourke, the Marchioness of Stafford and Lady Sybil St. Clair Erskine, the Marchioness of Bath and Lady Beatrice Thynne, the Marquis of Cholmondeley, the Marquese and Marquesa de Santurce and Lady Wilham Nevill, Countess Spencer, Countess Granville and Lady Victoria Leveson-Gower, Countess de Grey, the Earl of Dunraven, the Countess of Lathom and Lady Edith Wilbraham, the Countess of Enniskillen and Lady Christian Cole, the Earl of Ilchester, the Earl of Durham and Lady Anne Lambton, the Earl of Dalkeith and Lord Ernest Scott, the Countess of Sefton and Lady Rose Molyneux, Countess Howe and the Ladies Curzon, the Countess of Normanton and Lady Mary Agar and Miss Mary Liddell, Earl Craven, Viscount Clifden, Viscount Ennismore and Lady Beatrice Hare, Viscount Coke and the Hon. Violet White, Viscount and Viscountess Castlerosse and the Hon. Susan Baring, Viscountess Newport and the Hon. Miss Bruce, Viscount and Viscountess Melgund, Viscount and Viscountess Curzon, Viscount and Lady Violet Dangan and Lady Eva Wellesley, Viscount Dungarvan, Viscount Chelsea, Viscount Royston, Lady Hastings, Lord Capell, Lord and Lady Willoughby de Eresby and Hon. Margaret Willoughby, Lady Cloncurry and Hon. Miss Winn, Lord and Lady Brooke, Lady Wimborne and Hon. Miss Guest, Lord Sandhurst, Lord and Lady Edmund Talbot, Lord and Lady Alwyne Compton and Miss Gurney, Lord and Lady Chesham, Lady Rossmore and Miss Naylor, Lord Kenyon, Lord Garioch, Lady Alice Portal, Lady Margaret Levett and the Ladies Ashley, Lady W. Nevill, Lord Paulet, Lady Charles Beresford, Lady Forbes and Misses Forbes, Lady Claud Hamilton, Lord Carlow, Lady Mary Mills, Lord Lurgan, Lord Henry Vane Tempest, Lord and Lady Algernon Gordon Lennox, Lord Richard Nevill, the Hon. W. Coventry, the Hon. G. Joliffe, the Hon. Schomberg M'Donnell, the Hon. H. Coventry, Captain the Hon. E. Dawson, the Hon. Alistair and Mrs. Hay, the Hon. Mrs. Grosvenor, [[Social Victorians/People/Bourke|the Hon. Algernon and Mrs. Bourke]], the Hon. F. Dawnay, the Hon. E. J. Mills, the Hon. A. Johnstone, the Hon. Claude Hay, the Hon. Sidney Greville, the Hon. Mr. Dundas and Lady Hilda Dundas, the Hon. Thomas and Lady Idina Brassey and the Hon. Muriel Brassey, the Hon. Mr. Mansfield, Colonel the Hon. Henry Byng and Miss Byng, the Hon. Mrs. Oliphant, the Hon. Mr. Harbord, the Right Hon. Henry Chaplin, M.P., the Hon. G. Somerset, Sir William Gordon Cumming, Sir George Arthur, Sir Hubert Miller, Sir Charles Forbes, Sir William and Lady Eden, Sir George Chetwynd, Sir Frederick Johnstone, Sir John D. Poynder, Colonel and Lady Emma Crichton, the Hon. Miss Harbord, Captain and Lady Jane Combe, Colonel Paget, Captain Williams, Captain Ford, Captain Doyle, Captain Hanbury, Captain Campbell, Captain Constable, Captain Kinloch, Captain Longfield, M. Gutmansthal, Mrs. Bingham, Mr. Guthrie, Mrs. Arthur Paget, Mrs. Charles Wilson and Miss Wilson, Mr. Hulse, M.P., and Mrs. Hulse, Mr. de Murrieta, Mrs. Bischoffsheim, Mr. Spencer Portal, Mr. and Mrs. Gerald Paget, Mr. Leigh and Miss Leigh, Mr. Percy Daniell, Mr. Lister, Mr. Frederick Vincent, Mr. Cazalet, Mr. Alfred de Rothschild, Mr. John Savile Lumley, Mr. and Mrs Leopold de Rothschild, Mr. and Mrs. Skeffington Smith and Miss Taylor, Mrs. George Cavendish Bentinck, Mr. Alfred Farquhar, [[Social Victorians/People/Fanny Ronalds|Mr. Reginald Ronalds]], Miss Alice Ellice, Mr. George Marjoribanks, Mr. Wyndham, Mr. Dugdale, Mr. Montagu, Mrs. Sneyd, Mr. Wynn, Mr. Edward Hamilton, Mr. Murray Guthrie, Mr. George Lambert, Mrs. Sanford, Miss Mabel Forbes. Mr. Herbert Magniac, Mr. Bailey, Mr. Milner, Mr. Charles Kindersley, Mr. Beaumont, Mr. Walter Hervey, Mr. H. Hoare, Mr. Foley, Mr. Scott Murray, Mr. Coventry, Mr. Bertie, Mr. Jarvis, Mr. Guthrie, Mr. George Cavendish Bentinck, Mr. E. Charteris, Mr. Cholmondeley, Mr. Oswald Magniac, Mr. Cecil Campbell, Mr. and Mrs. Henry Oppenheim, Mr. Peploe, Mr. A. Charteris, Mr. Seymour Gore, Mr. Robert Antrobus, Mr. Elliot, M. de Champs, Mr. Wombwell, Mr. Charles Hoare, M.P., and many others.<ref>"Lord Alington's Ball." ''Morning Post'' 25 July 1890, Friday: 3 [of 8], Col. 5 b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900725/014/0003. Same print title and p.</ref></blockquote> ==August 1890== ===6 August 1890, Friday=== ==== Beautiful Women ==== The ''Lancaster Observer and Morecambe Chronicle'' reprinted a story from ''Vanity Fair'' about "Beautiful Women":<blockquote>Beautiful Women. — ''Vanity Fair'' asks what country in the world can show a more beautiful bevy of women than England? When Lady Helen Vincent, with her sister, Lady Cynthia Graham; when Lady Warwick with her sisters, the Duchess of Sutherland and Lady Westmorland; and when Lady Mar and Kellie, Lady de Trafford, Princess Pless, Lady de Gray[,] Lady Naylor-Leyland, the Duchess of Portland, Lady Alington, Lady Sophie Scott, Lady Chelsea, [[Social Victorians/People/Bourke|Lady Algernon Bourke]], Lady Henry Bentinck, Mrs. Willie Grenfell, Mrs. Rupert Beckett, Mrs. Arthur Paget, and Countess Clary are gathered together it is a sight to make an old man young; while Mrs. Gerard Leigh and her sister, Lady Yarborough, and Lady Powis, Lady Norreys, Mrs Farquharson, Mrs. Willie James, Lady Dalkeith, Mrs. Pretyman, Mrs. Gervase Beckett, Lady Rossmore, Lady Sarah Wilson, Lady Colebrook, Lady Hartopp, Mrs. Willie Walker, and Lady Milton are almost equally admired. The many pretty girls include Miss Enid and Miss Muriel Wilson, who still head the list of the unmarried beauties, Lady Beatrice Butler, Lady Alexandra Hamilton, Miss Kennard, Miss Madeline Bourke, and Miss Leila Crichton; while among the debutantes Lady Evelyn Crichton and Lady Vivian's daughters have had a great success, and Lady Alice Montagu, petite and charmingly pretty, has been so much admired. The daughters of Lady Gosford, Lady Verulam, Lady Leitrim, Lady Inchiquin, and Lady Grey have also been admired, as well as Lady Helen Stewart, Lady Katherine Scott, and Miss Chaplin.<ref>"Beautiful Women." ''Lancaster Observer and Morecambe Chronicle'' 6 August 1897, Friday: 2 [of 8], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001804/18970806/023/0002. Same print title and p.</ref></blockquote> ===25 August 1890, Monday=== Summer Bank Holiday ==September 1890== ==October 1890== ===22 October 1890, Wednesday=== Arnold Dolmetsch went to the Reading Room (Campbell 22). ===25 October 1890, Saturday=== [[Social Victorians/Loder De Vere Beauclerk Wedding|Loder—De Vere Beauclerk Wedding]] ===31 October 1890, Friday=== Halloween. ==November 1890== ===5 November 1890, Wednesday=== Guy Fawkes Day ===19 November 1890, Wednesday=== Arnold Dolmetsch, with some of his students and his daughter Helene, provided musical illustrations for a lecture given by Dr. Frederick Bridge on "Mozart as Teacher," probably at Gresham College, where Bridges was professor (Campbell 34). ===21 November 1890, Friday=== Dr. Frederick Bridge's last fall 1890 lecture at Gresham College, for which Dolmetsch provided illustration. Campbell says, "it is, in fact, the first record in modern times of music by Byrd, Dr. Bull, Jenkins, Simpson, Lawes, Locke and Purcell being played 'on the instruments for which it was written'" (Campbell 34). ===25 November 1890, Tuesday=== Arnold Dolmetsch got a three-month pass to the Reading Room and "thereafter held yearly tickets until 1916" (Campbell 22). ==== 1890 November 29, Saturday ==== ===== [[Social Victorians/Dudley-Beckwith Wedding 1890-11-29|Wedding of Francis Dudley, Lord Leigh and Héléne Forbes Beckwith]] ===== ==December 1890== Sometimes in December 1890, [[Social Victorians/People/William Robert Woodman|William Woodman]], one of the founding members of the [[Social Victorians/Golden Dawn|Golden Dawn]], died (Gilbert 86 12). === 4 December 1890, Thursday === ==== [[Mure-Portal Wedding 1890-12-04|Wedding of Miss Mure and Mr. S. J. Portal]] ==== ===19 December 1890, Friday=== "The eleventh annual ''Truth'' Toy Exhibition for the children in London hospitals, workhouses, and infirmaries was held at the Grosvenor Gallery, Bond-street, on Dec. 19 and 20. There are over 22,000 children in these institutions, and each of them has a separate toy, besides the large and more expensive toys which are presented for the general use of the inmates. A special feature of the show was the dolls dressed by lady readers of ''Truth''. Several special donations in connection with the fund have been made, the principal of which is that of 10,200 new sixpences sent for distribution by the same liberal donor who has given a similar present on former occasions; while Mr. Tom Smith has had specially made, and has forwarded for distribution, over 22,000 crackers; and Mr. Horniman (as stated in our last issue) has given to the adults in hospitals and workhouses 1200 packets of tea, to be given to them as Christmas boxes." (Illustrated London News (London, England), Saturday, December 27, 1890; pg. 806; Issue 2697, Col. B) ===20 December 1890, Saturday=== "In order that the adults connected with the hospitals and workhouses may participate in the good things given to the children in the ''Truth'' toy distribution, Mr. F. J. Horniman, of Wormwood-street, City, has sent Mr. Labouchere 1200 packets of tea to be given to them as a Christmas box." (Illustrated London News (London, England), Saturday, December 20, 1890; pg. 774; Issue 2696, Col. B) From the ''Times'', 22 December 1890, "The Late Sir Edgar Boehm": <blockquote>On Saturday, in accordance with the special request of the Queen, the late Sir Edgar Boehm, R.A., was buried in St. Paul’s Cathedral, in the south aisle of the crypt, the spot being known as “The Painters Corner.” [new paragraph] The funeral cortége left the residence of the deceased at about 11 o’clock, there being five mourning and several private carriages. In the first carriage were the chief mourners, Sir Edgar Collins Boehm, the only son of the late baronet, and Mr. Conrad Herapath. The second and third carriages were occupied by Sir Frederic Leighton, P.R.A., Sir Nigel Kingscote, Mr. Richard Mills, Colonel Francis Baring, Mr. Alma-Tadema, R.A., Sir R. H. Collins, Mr. Edward J. Poynter, R.A., and Mr. Duncan MacGregor, all of whom subsequently, at the cathedral, acted as pall bearers. The fourth carriage contained representatives of the Council of the Royal Academy – Mr. Edwin Long, R.A., Mr. J. B. Burgess, R.A., Mr. F. Goodall, R.A., and Mr. Beavis. In the fifth carriage were the following members of the professional staff of the deceased – Mr. R. Glassby, Mr. E. Lantéri, Signor Finili, and Herr Gross. The cortége stopped for a few moments at the Royal Academy, in order that it might be joined by two carriages containing other representatives of the institution. Leaving Piccadilly, the route taken was Waterloo-place, Pall-mall, Cockspur-street, the Strand, Fleet-street, and Ludgate-hill, the cathedral being reached about noon. The procession was met at the great west door by Canon Gregory, the Archdeacon of London, the Rev. Dr. Baker, Minor Canons Milman, Russell, and Kelly, and the choir, who had a few minutes previously left the vestry; by Major Bigge, Equerry to the Queen, representing her Majesty; Colonel Stanley Clarke, who attended on behalf of the Prince and Princess of Wales; and Colonel W. J. Colville, representing the Duke of Edinburgh. The [[Social Victorians/People/Princess Louise|Princess Louise]] (Marchioness of Lorne) attired in deep mourning, and accompanied by [[Social Victorians/People/Arthur Collins|Lieut.-Col. Arthur Collins]] and Lady Sophia Macnamara, arrived at the cathedral a few minutes before 12 o’clock, and was conducted to a seat in the south side of the nave, close to the late Sir Edgar Boehm’s two daughters, upon whom her Royal Highness had called earlier in the morning. As the procession came slowly up the nave towards the chancel, “I am the resurrection of life” and other sentences of the burial service were chanted to music by Dr. Croft. The coffin, which was completely hidden by splendid wreaths, was preceded by the choristers and the clergy. Then followed Sir E. C. Boehm, with Mr. Conrad Herapath and Mr. Alfred Gilbert, A.R.A., the officers and council of the Royal Academy, and members of the deceased’s studio. The coffin was deposited upon a bier, which had been placed under the dome in front of the opening into the crypt, through which the body was subsequently lowered. The mourners having been conducted to the seats which had been reserved for them, and the clergy and choir having proceeded to their places in the chancel, Psalms xxxix. and xc. Were sung to music by Purcell and Felton, followed by Spohr’s anthem, “Blest are the departed,” and the lesson 1 Cor. Xv. 20, which was read by Canon Gregory. Whilst the hymn “Days and moments quickly flying” was being sung the Archdeacon of London and Minor Canon Milman proceeded from the chancel and took up positions by the side of the coffin. The remainder of the service preceding and following the lowering of the body into the crypt was read by Minor Canon Milman, and the collect was impressively recited by Archdeacon Sinclair. The late Dr. Newman’s beautiful hymn, “Lead kindly light,” was afterwards sung by the choir, and the service was brought to a close by Beethoven’s Funeral March, which was played in compliance with the special request of her Majesty, at whose desire also Chopin’s Funeral March had preceded the service. Dr. Martin was the organist. While Beethoven’s march was being played the son and daughters of the late “Sculptor in ordinary to her Majesty,” together with other mourners, descended the crypt by the door near Howard’s monument, and saw the coffin for the last time. It bore the following inscription:- “Joseph Edgar Boehm. Born July 6, 1834; died at his work December 12, 1890. Thy will be done.” The coffin had already been deposited in the brick grave prepared for it, which is next to that of Sir Edwin Landseer. Immediately around it lie the remains of painters and sculptors of bygone days – Sir Joshua Reynolds, James Barry, Benjamin West, Henry Fuseli, Sir Thomas Lawrence, J. M. W. Turner, and John Henry Foley, while within half-a-dozen yards is the tomb of the great architect of the cathedral, Sir Christopher Wren. The coffin as last seen was covered by the floral tributes of loving friends, the symbolic laurel wreath forwarded by her Majesty surmounting them. This bore the inscription, “A tribute of gratitude for many beautiful memorial works executed for her. From Victoria, R. I.” Attached to the beautiful wreath sent by the Prince and Princess of Wales were the words, “A token of sincere regard and friendship.” Wreaths were forwarded by other members of the Royal Family, by Sir John and Lady Millais, Lord and Lady Reay, the Countess Sydney, Lord and Lady de Vesci, Mr. Henry Irving, Mr. and Mrs. Bancroft, by members of the council, students and officers of the Royal Academy, the Society of Artistic Sculptors (Vienna), and by the London Hungarian Association." (http://glassby.com/Chapter1.htm) </blockquote> ===24 December 1890, Wednesday=== "The people of South London and adjacent suburbs have to be grateful to Mr. Frederick Horniman, a well-known City merchant residing at Dulwich, for a munificent and beneficial Christmas gift. This gentleman has, during thirty years, formed a collection, worth above £100,000, of specimens of art, European, Indian, Chinese, and Japanese, also of decorative manufactures, and of ethnology and natural history, and rare books and manuscripts, at Surrey House, Forest Hill, which he intends to present to the public. The museum was opened by Sir Morell Mackenzie on Wednesday afternoon, Dec. 24. It already comprises twenty-four rooms, containing many beautiful specimens of horse and chain armour, Elizabethan bed-room furniture, an old English parlour and pantry, an Oriental armoury, a bible and manuscript room, an Egyptian gallery, and ethnographical saloon, a book and embroidery saloon, a porcelain and glass room, and a zoological saloon, with two live Russian bears and a monkey. We may describe the museum further, after another visit. Our Artist has sketched, in the ethnograhical saloon, an idol's shrine, made of cocoanut-fibre, from Fiji. Mr. Horniman, in acknowledging a vote of thanks, said that in an adjoining house he had accumulated a vast library of bibles and entomological works and specimens. It was his intention to build a hall at the back of the present building to contain a thousand persons, while the galleries would contain a library, and a stage would be provided, so that both eye and ear could be amused. He also contemplated buying sufficient land to enable a public body to erect class-rooms and gymnasia. Three cheers were given for Mr. Horniman, and the company proceeded to inspect the museum." "(Mr. F. Horniman's Museum, Forest Hill." Illustrated London News (London, England), Saturday, January 03, 1891; pg. 7; Issue 2698, Col. C) ===25 December 1890, Thursday=== Christmas Day ===26 December 1890, Friday=== Boxing Day ==Works Cited== *"Besant Chronology.pdf." From Anderson, Nancy Fix, Walter L Arnstein, Deborah Logan, and Susie L Steinbach, Volume Editors. Lives of Victorian Political Figures, Part III: Queen Victoria, Florence Nightingale, Annie Besant and Millicent Garrett Fawcett by their Contemporaries. Volume 3 of Lives of Victorian Political Figures. Series Editors, Nancy LoPatin-Lummis and Michael Partridge. London: Pickering and Chatto, 2008. *Crawford, Elizabeth. The Women’s Suffrage Movement: A Reference Guide, 1866–1928. Google Books. Accessed 14 October 2010. hcyags62viw8sz0jn6ec9oz2p3kr22f 2835587 2835585 2026-10-02T23:07:37Z Scogdill 1331941 /* February 1890 */ 2835587 wikitext text/x-wiki [[Social Victorians/Timeline/1840s|1840s]] [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] 1890 [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] [[Social Victorians/Timeline/1896 | 1896]] [[Social Victorians/Timeline/1897 | 1897]] [[Social Victorians/Timeline/1898 | 1898]] [[Social Victorians/Timeline/1899 | 1899]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]] ==January 1890== Sometime in January 1890, [[Social Victorians/People/Horniman|Annie Horniman]] was initiated into the Golden Dawn (Gilbert 86 144). Sometime in 1890 Annie Besant and [[Social Victorians/People/Helena Blavatsky|Helena Blavatsky]] founded the Working Women's Club in the East End (Besant Chronology.pdf). Maye Dilke was a member (Crawford 170). [[Social Victorians/People/Frank Harris|Frank Harris]] says,<blockquote>One day in 1890 I had George Meredith, Walter Pater and [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] dining with me in Park Lane and the time of sex-awakening was discussed. Both Pater and Wilde spoke of it as a sign of puberty. Pater thought it began about thirteen or fourteen and Wilde to my amazement set it as late as sixteen. Meredith alone was inclined to put it earlier.<ref>Harris, Frank. ''My Life and Loves''.</ref></blockquote> ===1 January 1890, Wednesday, New Year's Day=== <blockquote>Sharp frost was experienced over the East of England yesterday morning, and in South London the thermometer had been as low as 27deg., and 28deg. in Hyde-park. It rose during the day to 41deg., but in the evening frost again set in. Later the thermometer rose again to 32deg., and there was an appearance as of coming rain.<ref><cite>The Morning Post. 1 January 1890 (No. 36,675): P. 4, Col. F.</cite></ref></blockquote>Influenza is spreading all over Europe. According to the "Court Circular: Orders for Court Mourning" in the Morning Post,<blockquote>A second supplement to the Gazette, issued last night, contains the following: — "Lord Chamberlain's Office, January 1. "Orders for the Court's going into mourning on Thursday next, the 2d instant, for her late Majesty the Empress of Brazil, viz." — "The ladies to wear black dresses, white gloves, black or white shoes, feathers, and fans, pearls, diamonds, or plain gold or silver ornaments. The gentlemen to wear black Court dress, with black swords and buckles. "The Court to change the mourning on Thursday, the 16th instant, viz.: — [new paragraph] The ladies to wear black dresses, with coloured ribbons, flowers, feathers, and ornaments, or grey or white dresses, with black ribbons, flowers, feathers, and ornaments. The gentlemen to continue the same mourning. "And on Thursday, the 23d instant, the Court to go out of mourning."<ref><cite>"Court Circular: Orders for Court Mourning." The Morning Post, 2 January 1890 (No. 36,676): page 5 Col. G.</cite></ref></blockquote> ===9 January 1890, Thursday=== ==== The York Hunt Ball ==== This account is from the ''Irish Society'', which explains which women's costumes are the focus:<blockquote>The fashionable world and his wife are out of town just now, having good times at country houses. They shed the light of their brilliant presence on those annual functions known as county balls, and enjoy themselves also at those popular and picturesque entertainments to which they are bidden by the masters and members of the various hunts. One of the smartest dances of last week took place at York on Thursday evening. It was in fancy costume, and was in aid of funds for the York County Hospital. Once upon a time the journey from London to the Minster City of the North was a far cry, and one not lightly undertaken. Now it is luxurious and short. In consequence this hall presented the appearance of one in town in mid-season. There was conspicuous for height and handsomeness the familiar figure and face of Countess de Grey. Lady Helen Duncombe, too, who shares with her Grace of Leinster, and Lady Cynthia Graham, the family beauty of the House of Feversham, was seen in the marvellously becoming (to her) costume of the Empire period. Lady Lilian Wemyss, a lady "divinely tall,” and appropriately fair, was garbed in pink and white of a fashion hailing from the Empire. A "square;” in which the lady dancers were all gowned empirically, was the event of the evening. An event worth remembering it was. The costumes were superb, but their attractions were nowhere in comparison with the charms the [Col. 1c–2a] wearers. These were all tall and each one handsome. Lady de Trafford’s pretty face and fine figure showed to great advantage in her gown of white satin, veiled with gold-bordered ''chiffon'', and with an Empire trained cloak from her shoulders of Rose du Barri velvet lined with white satin, and bordered with gold galon. The colour of the velvet in what is seen in the half-yellow, half-scarlet petals of the soft textured nasturtium bloom. [[Social Victorians/People/Bourke|Mrs Algernon Bourke]] looked a picture in a Gainsborough gown. The white satin skirt was flounced with sable and veiled with ''chiffon'', the setuage of which was left to show without being hemmed up. There was a broad sash of rose-pink silk and each buttonhole was filled round with crimped lisse. One might go on for long enough describing costume after costume, and then leave them half out. Therefore, we shall, like the judge "sum up” and "charge” for a verdict of the utmost brilliancy.<ref>"Our London Letter." ''Irish Society'' (Dublin) 11 January 1890, Saturday: 17 [of 24], Col. 1b–2a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001939/18900111/042/0017. Same print title, p. 29.</ref></blockquote> ===16 January 1890, Thursday=== <blockquote>About two hundred members of the newly formed Dulwich Literary and Scientific Association attended the conversazione given on Jan. 16 at Surrey House Museum, Forest Hill, when Mr. Horniman, F.R.G.S., acted as cicerone round the rooms, richly stored with historical relics, ornithological curiosities, and art treasures.<ref>''Illustrated London News'' (London, England), Saturday, January 25, 1890; pg. 120; Issue 2649, Col. 3.</ref></blockquote> === 1890 January 28, Tuesday === According to the ''Dundee Evening Telegraph'' on 28 January 1890 people were buying "Exploration" shares, which funded exploration expeditions with the possibility of rights to minerals or resources or the development of monopolies for goods that were "discovered."<blockquote>Royalty and the peerage have gone in for "Exploration" shares. The Duchess of Albany is the possessor of fifty shares, which hardly seems to assume the dimensions of a royal investment. Certainly the general public would incline to the idea that a princess would never condescend to invest so small a sum as £25 in anything unless it were a hat or a jacket. The Duke of Fife, on the other hand, has been allotted a thousand shares. The Duchess of Manchester and the Countess of Mandeville have thirty each.<ref>"Notes — Mainly Personal." ''Dundee Evening Telegraph'' 28 January 1890, Tuesday: 2 [of 4], Col. 2b [of my 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18900128&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''Evening Telegraph'', n.p.</ref></blockquote> === 1890 January 29, Wednesday === [[Social Victorians/Santurce Murrieta Wedding 1890-01-29|Santurce Murrieta Wedding 1890-01-29]] === '''1890 January 30, Thursday, through at least''' 1890 February 8, Saturday === Kim, Lord Mandeville was in Brighton — at 80, King's-road — at least as early as 30 January.<ref>"Brighton Visitors." ''Brighton Gazette'' 30 January 1890, Thursday: 3 [of 8], Col. 2a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000938%2F18900130&page=3&article=073&stringtohighlight=mandeville+viscount. Print title: ''Brighton Gazette and Sussex Telegraph'', p. 3.</ref> The ''Lady's Pictorial'' lists "recent arrivals" in Brighton:<blockquote>Though a good many people have left Brighton, still fresh visitors arrive, and the front is crowded when the weather is a little kind. Amongst the most recent arrivals may be mentioned the Countess of Strafford, Lady Susan Byng, Lady Chelmsford, Lord and Lady Seaton, Lady Jane Grimston, [[Social Victorians/People/Mandeville|Lord Mandeville]], Lord Romilly and the Hon. John Romilly, Viscount and Viscountess Hampden, the Hon. R. Somerset, the Hon.  Thomas Dundas, the Hon. Sidney Greville, the Hon. A. W. Verney Cave, Lady Elizabeth Cartwright, Colonel Sir Alfred and Lady Kirby, Mr. Bazley White, M.P., and Lady Bazley White, Sir Hugh Montgomery, Sir Frederick and Lady Perkins, Lady Ellis, General Seymour, &c., &c.<ref>Dorothy. "Brighton Notes." ''Lady's Pictorial'' 8 February 1890, Saturday: 51 [of 60], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005980%2F18900208&page=51&article=671&stringtohighlight=mandeville+viscount. Same print title, p. 203.</ref></blockquote> ==February 1890== === 1890 February 6, Thursday === Before 6 February but not long before. The prize-fighter Pete Jackson fought a match at the New Pelican Club.<blockquote>It is always well to hear that distinguished visitors have enjoyed the hospitality which we have extended to them. This satisfaction is afforded us in full measure by the eminent "Parson" Davis, manager of Pete Jackson, the prize-fighter, who has just been regaling an American interviewer with an account of his experiences in England :— He had been treated in splendid style, and Jackson was received well everywhere. Nothing would give him greater pleasure than to revisit England, and he hoped soon to do so. "I shook a paw with the Prince,” said the parson. "We had a high old time with Albert E. His Royal Highness was exceptionally friendly with Jackson, and actually made a pump handle of his arm. He shook hands with him again and again. The Prince is in capital  health, and" —— But at this point the interviewer mysteriously breaks off. The Prince, however, was not the only British notable to whom Parson Davis referred. He explained that the cause of the trouble with Smith at the New Pelican Club was that the gloves furnished Jackson were too small in the fingers, and when he closed his hand he almost split them, and could not fight. The matter was explained to the officers of the Club, where the fight took place, and other gloves were procured for Jackson :— Fair play was given, and the audience, which included [[Social Victorians/People/Mandeville|Lords Mandeville]] and Lonsdale, were loud in their admiration of Jackson's skill.<ref>"The Tittle Tattle of the Week." ''Pall Mall Budget'' 6 February 1890, Thursday: 4 [of 36], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005967%2F18900206&page=4. Same print title, p. 162.</ref></blockquote>Possibly related is this, from 24 January 1890:<blockquote>Further details (says the London ''Evening'' ''Post'') have transpired with regard to the fracas at the Tracadero between Charley Mitchell and Mr Abington. It seems that the cause of the row was the conspicuous and irritating production of Smith’s colours by a gentleman in the stalls in front of Mitchell. These acted upon him like the proverbial red rag, and turning to Mr Abington, he said — "You have got this up for me. I will put this cigar in your eye." Here Lord Mandeville acted the part of the peacemaker, and an adjournment was made to the bar, where the real row took place. Notwithstanding what the ''Sportsman'' says, it is a fact that "the Squire" did take off his coat to Mitchell.<ref>"The Laird of Strichen and the Prize Ring." <i>Fraserburgh Advertiser</i> 24 January 1890, Friday: 5 [of 8], Col. 5a [of 5]. <i>British Newspaper Archive</i> https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001470%2F18900124&page=5&article=035&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref></blockquote> === 12 February 1890, Wednesday === ==== Lady Constance Leslie's Reception ==== <blockquote>On Wednesday last Lady Constance Leslie had an evening party at her residence in Stratford-place. The numerous guests included Prince Ernest of Hohenlohe, Princess Victor of Hohenlohe, and the Countesses Gleichen, the Austro-Hungarian Ambassador, Count Mensdorf, Count and Countess Henry Lützow, Mr. and Mrs. Henry White, Baroness de Brienen and Mdlles de Brienen, the Duke of Abercorn, the Duchess of Cleveland, the Marchioness of Blandford and Lady Frances Spencer Churchill, Earl and Countess of Portarlington, Earl Stanhope, the Earl of Northbrook, the Earl and Countess of Arran, the Earl and Countess of Erne and the Ladies Cole, Julia Countess of Jersey and Mr. Brandling, Viscount Clifden, Viscountess Bary, Viscount Torrington, Viscount and Viscountess Castlerosse and Lady Louisa Beauclerk, Viscount and Viscountess de Vesci and Lady Sarah Spencer, Viscountess Galway, and Mr Caryl Ramsden, Lord de L'Isle and Dudley, Lord Alcester, Lord and Lady Wantage, Lord Ashbourne, Lady Henry Grosvenor and Lady G. Ogilvy, Lady Cloncurry, Lord Morris and Mr. M. Morris, Lady Charles Beresford, Lord Rowton, Lady Blanche Hozier, Lady Colville, Lady Rayleigh, Lady Constance Lytton and Mr. Charles Earle, Lady Evelyn Sutton, Lady Ernest Seymour and Ladies Fortescue, Lord and Lady Sandhurst, Lady Mary Lloyd, Lady Tweedmouth, Lady Windsor, Lord and Lady William Seymour, Lady Guy Campbell, Lady Egerton, Lady Fitzgerald, Baron d'Erlanger and Mr. Charles Derenberg, Hon. R. C. Spencer, M.P., and Mrs. Spencer, Hon. Evelyn Ashley, Hon. W. Mansfield, Hon. Lionel Ashley, Hon. A. Hood, Hon. William Lowther and Miss Lowther, Hon. Sydney Greville, Hon. Fitzroy and Mrs. Stewart, Hon. Frederick Leveson Gower, Hon. Kenneth Howard, Hon. Francis Villiers, Hon. Francis Bertie, Hon. Arthur Walsh, Colonel the Hon. Charles Edgcumbe, General Hon. John Bourke, General and Hon. Mrs. Ellis and Mrs. R. Sneyd and Miss Ellis, [[Social Victorians/People/Bourke|Hon. Algernon and Mrs. Bourke]], Hon. Alan and Hon. Evan Charteris, Hon. Alexander Yorke, Hon. Derek Keppel, General Sir Dighton Probyn, Hon. Percy Wyndham, Hon. Everard Baring and Hon. Susan Baring, Hon. Mrs. Brett, Hon. Claud Hay, the Right Hon. Henry Matthews, Sir George Arthur, Sir George Maude, Sir Philip Currie, C.B., Sir Redvers and Lady Audrey Buller, Sir Charles Grant, Sir Mackenzie Wallace, Sir Herbert Maxwell, M.P., Sir Oscar Clayton, Sir Robert Gresley, Sir Algernon West and Mr. Horace West, Sir George and Lady Russell and Miss Brackenbury, Sir Henry and Lady Longley, the Sub-Dean of the Chapels Royal, General Crealock, General Charles Fraser, Colonel Saunderson, M.P., and Hon. Mrs. Saunderson, Colonel and Mrs. Stewart and Miss Streatfeild [siv], Colonel and Miss Balfour, Colonel and Mrs. Stanley Clarke and Miss Clarke, Colonel R. Baring, Captain and Mrs. Jekyll, Captain and Mrs. J. Bagot, Captain Lambert Ward, Captain A. Bagot, Mr. Charles Hall, Q.C, M.P., Mr[.] Hussey Walsh, M.P., Mr and Mrs. W. Hartpole Lecky, Mr. and Mrs. Richard Combe, Mr Henry Foley, Mrs. Arbuthnot and Miss Guthrie, Mr. and Mrs. Heseltine and Miss Dorothy Heseltine and Lord Cantelupe, the Miss Stopford Brooke, Mr. and Mrs. Thackeray Ritchie, Mr Edmund Byng, Mr. and Mrs. Wilton Phipps, Professor Sidney Colvin, Mr. and Miss Tennant, Mr. Godfrey Webb, Mr. and Mrs. Turton and Mr. Sidney Ponsonby, Mrs. Adare, Mrs Cecil Paget and Mr. Hugo Wemyss, Mrs. L. Drummond, Mr. and Mrs E. Hope, Mrs. L. Seymour and Miss Seymour, Mr. Guy Repton, Mr. and Mrs. Julian Sturgis and Miss Beresford, Mr. Poynter, R.A., Mrs. Daniel Cooper, Mr. Hamilton Aïdé, Mr. Alwyne Maude, Mrs. C. Bentinck, Mrs. Bingham Mildmay, Mr. Philip Habord, Mr. and Mrs. George Bulteel, Mrs. Beerbohm Tree and Miss Julia Neilson, Mr. Hallé [Halle?], Mr. Rudolph Lehmann and Miss Lehmann, [[Social Victorians/People/Fanny Ronalds|Mr. R. Ronalds]], Mr. Ellice, Mr. Arthur, Madame Blumenthal, Miss Julia Leslie, Mr. and Mrs. Jeune, Mr and Mrs. Charles Stuart-Wortley and Misses Millais, Mrs. Arkwright, Mr Joachim, Mr. Gillett, Mr. Pulteney (Scots Guards), Colone| Cuthbert Larking, Mr. Harry Higgins, M. Tividar Nachez, Mr. G. Ross, Herr Johannes Woolff, Mrs. Charles Bagot and Miss Bagot, Mrs. Bischoffsheim, Mr. Henry Petre, Miss Balch, Mr. and Lady Maud Ramsden, Mr. Adolphus Liddell, Mr Knowles, Mr. Grenfell, Mr. Edward Hamilton, Mrs. Henry Gordon and Miss Gordon, Mr. Henry James, Mr. St. Leger, Mrs. Dacre Hamilton and the Misses Nugent, Mr. Harbord, Mr Somerset, Mr. Ruggles-Brise, Mr. and Miss Hughes of Kinmel, Mr. and Mrs Douglas Freshfield, Mrs. Homer, Mr. W. H. Mallock, Mr. and Mrs. Robert Benson, Mr. H. Seymour, Mr. Ernest Law, Mr. Hugh Grimston, Mr. Lionel Cust, Mr. Chappell, Mr. and Mrs. Du Maurier, Mr. Boughton, R.A., and Mrs. Boughton, Mr. R. Roche, Mr. Montagu Wilson, Mr. Herbert Hope, &c.<ref>"Lady Constance Leslie's Reception." ''Morning Post'' 15 February 1890, Saturday: 2 [of 8], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900215/008/0002. Same print title and p.</ref></blockquote> === 24 February 1890, Monday === Lilly Langtry starred in Shakespeare's ''As You Like It'' at the St. James's Theatre. The Daily Telegraph had a long review and mentioned more names:<blockquote> The motto of Petronius, which is said to have adorned the Globe Theatre in Shakespeare's time, "Totus mundus agit histrionem," and which is reproduced in Jaques' famous speech, may be said to have received a novel illustration in the brilliant audience which completely filled the St. James's Theatre last evening. A very distinguished section of the world "played the actor" in the stalls, boxes, and dress-circle, and evinced the keenest dramatic interest in the long-deferred production of "As You Like It." The Prince and Princess of Wales, with the Duchess of Fife, surveyed the scene from the Royal box, while amongst the many who secured seats for the first were the Austrian Ambassador, the [[Social Victorians/People/Louisa Montagu Cavendish|Duchess of Manchester]], Countess de Grey, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lady Colin Campbell, Lord Wolverton, Lord and Lady Coleridge, Sir Charles Russell, Sir Edward and Lady Guinness, Baroness de Stern, and Mr. Alfred Rothschild. Naturally enough there was much curiosity in seeing once more on the London stage an actress who has since her last appearance devoted so much serious study to her profession, and gained so much success both before American and provincial audiences; while the sympathy evoked by her recent severe illness, delaying for four weeks the production of the comedy, added an almost pathetic note to the general excitement and enthusiasm. In the second scene of the first act, when the tenderly attached cousins make their earliest appearance on the stage, Celia's first words, "| pray thee, Rosalind, sweet, my coz, be merry," answered by Rosalind’s "Dear Celia, I show more mirth than I am mistress of," were almost drowned in the applause which accompanied the recognition of Mrs. Langtry. From time to time, whenever opportunity served, enthusiasm broke out anew, culminating in the triumph of plaudits which formed the fit ending to the gracefully-delivered epilogue with which Rosalind takes her farewell of her audience. There can be no doubt that, with all due allowance for the inevitable nervousness which accompanied a first production before a critical though sympathetic house, the revival of the immortal comedy was generally pronoanced to be a success. The fancifally-named "As You Like It" — where even in nomenclature the author confesses his indebtedness to the romance of Thomas Lodge — with greater dullness than is usually the case with Shakespeare's plays. Indeed interest is hardly aroused till the scene before the Duke's Palace, where Rosalind and Celia witness the wrestling — which after a few more performances will probably go less stagily than it did last night — between Charles and Orlando. It is only when we are transported to the Forest of Arden that Shakespeare's romantic fancy allows itself full course, and it was only when she was accoutred in the doublet and hose that Mrs. Langtry exercised her full charm upon the audience. The difficulty for the actress in the character of Rosalind is the contrast between the early abandon of her frankly-expressed love for the son of Rowland de Bois and the subsequent archness and reserve with which she plays with her lover in the forest. Orlando goes through precisely an opposite change. From the careless desperateness of a man who is out of favour with fortune in the opening scene, he changes to the sentimental devotion of a "love-shaked fancy-monger" in the sylvan glades. If that interchange of parts be missed much is lost in the sprightly  contrasts of the drama. Unfortunately, no sooner did Orlando enter on the wrestling-ground last night, than he started and caught at his heart in obvious suggestion of love at first sight. Much is lost of the utter wonderment at Rosalind’s unexpected graciousness in giving him her chain, which keeps him tongue-tied later, if he betrays so early the fatal ravages of Cupid — a betrayal which is hardly likely in a man whom neglect and misery have driven to the last stage of reckless hardihood. Rosalind must love her champion before her winning thanks for his prowess have made the starred younger brother first marvel at and then adore the frank and generous advances of his benefactress. In the fine words, "My father was no traitor," Mrs. Langtry rose to the full height of the character she was depicting. Throughout the greater part of the first act there was grace, loveableness, charm, but also a kind of flatness, or, at all events, an even and somewhat uninteresting level of characterisation which did not convey the promise of all that this loveliest creation in Shakespeare's gallery could do, when the Court life was exchanged for the freer and less conventional existence in the woods. All the more praise, therefore, is Mrs. Langtry’s due, that when she finds herself in the Forest of Arden, she brings before us a truer Rosalind. The beautiful first scene between the disguised Ganymede and her amorous swain showed Mrs. Langtry at her best. The quick and nimble wit, the archness and pretty coquetry of a girl who is sick at heart with love and feeds her passion in a playful and adorable tyrannising over the unconscious lover, were depicted with a reality which won instant recognition from the spectators. For the rest, however, Mrs. Langtry’s Rosalind was an unequal performance, now rising with unexpected force into the regions of pure comedy, now sinking to an ordinary level of  uncomfortably-disguised womanishness. There were bright patches of power in the midst of a somewhat dull and colourless expanse. Much of this was, no doubt, due to the excitement and fatigue of a first night, to which also may be attributed possibly the sameness of some of her gestures and a restlessness which was something more than Rosalind's anxiety to win in her own incomparable way the happiness of which she was in search. A few nights hence firmness of carriage and self-control of manner will fill in the details of a character which was most graphically and sympathetically sketched in outline at the St. James’s last night. Mr Laurence Cautley as Orlando made most favourable impression on his audience. Through all the earlier scenes — with the exception of that wholly unnecessary lapse into amorousness on the lawn of Duke Frederick, which has been already noted — he sustained the character of a picturesque and youthful gallant with considerable success, though, possibly, a stronger dash of manly independence might not have come amiss to the Shakespearean hero. Touchstone is a more difficult part to play. Indeed, in some respects, he is the most difficult of all Shakespeare's clowns. He is on the borderland between the earlier, more conventional style of Costard, Launce, and Launcelot, and the far more elaborate and important clowns in "Twelfth Night" and "King Lear." He is not so clever as Feste, not so human and real as that inimitable and tender-hearted Edgar who tries to solace the wild ravings of his discrowned king. He is halfway between such farcical beings as Kempe and Tarlton probably depicted, and that witty, self-conscious, irresponsible Jester, which Shakespeare ultimately made an entirely new creation on Elizabethan boards. It is no discredit to Mr. Sugden that he failed to realise so ingenious a conception. He presented a Touchstone who was full of amusing quips, and he went through all the catalogue of points in the Book of Honour in the fifth act with admirable skill; the Touchstone of "As You Like It" we are probably not likely to see often on the modern stage. Mr. Bourchier’s Jaques was a far higher piece of portraiture, grave and bitter and cynical as that early disciple of Schopenhauer should be, with touches of humour which only serve to show the depths of a naturally perverse and misanthropical nature. The celebrated "Seven Ages" speech  was capitally delivered, with a conscientious care and easy mastery which are rare enough in an actor who has so lately left the ranks of the amateurs. For the rest, the Adam of Mr Everill, and the Andrey of Miss Marion Lea seemed to give most pleasure to the audience, though she was sometimes — espectally in the pronunciation of the word "poetry" — inclined to represent the boorishness of the wench with an accent  which reproduced simulated inebriation rather than native clumsiness. Miss Armbruster as Hymen also deserves a word of praise. The piece was produced under the direction of the Hon. Lewis Wingfield, and the scenery, which in the Forest Glade of Acts II., III., and IV., formed a beautiful stage picture, was designed by Messrs. Perkins and Bruce Smith. After all, was any one ever wholly satisfied with the representaton of one of Shakespeare's comedies? No thing is further from the truth as a general statement than the pedant’s theory that Shakespeare is for the study, and not for the boards. If ever, however, the paradox be true that a great dramatic artist suffers when his plays gain a dramatic representation, it is when "As You Like It," "Twelfth Night" and "Midsummer Night's Dream" are brought face to face with a modern audience and amid nineteenth century surroundings. It ts not with impunity that the light that never was on sea and land is parodied by the glare of the footlights. For, honest stage-craftsman as Shakespeare was, in the full spring-tide of his poetic genius there was a divine and masterful intolerance of the real and the commonplace, a reckless flight into an ideal world, peopled by beings coined in the mint of his dainty imagination — aerial and disembodied creations whose actions and characters refuse to submit themselves to the narrow rules of the every-day world. In such a sphere banished lovers shall be predestined to meet and cement their happy unions, and lions and snakes shall be the habitual denizens of a forest where shepherds and shepherdesses prattle their erotic strains, and even a crafty Oliver shall repent of his past sins and fall in love with gentle Celia, owing to the poetic exigencies of Cloud Cuckoo Land. How can one  see all these things with the cynical eye of the modern ''Fin de Siècle'' and not feel an inevitable shock of dissatisfaction and surprise? How is it possible to avoid the jarring sense of discord when the dream picture is realised in the setting of prosaic reality? For Shakespeare's comedy is a romantic work of the mind, a refuge for all those delightful improbabilities which an indolent or whimsical fancy strings together to construct a series of complications and incidents which interest us, while they elude the sober tests of reason and judgment. But we must not carry this vein of thought too far. It is true that we have in such plays comedies of incident rather than of character. Yet Shakespeare is too and heart-whole to linger only at the Ivory gate. He throws even into these merry plots the force of that instinctive knowledge of human nature which produces real personages, tricked out in witty and imaginative guise. Shakespeare was able to pour everything into his comedies which he did into his tragedies, though in the one there is only suggestion, while in the other there is the elaboration of that is deep and tragic as well as humorous in life. Rosalid, Orlando, Touchstone, and Jaques are none the less real because they are ideal; and it is on these grounds that we can praise the conscientious and capable artists who presented these characters in flesh and blood on the St. James's stage. Nor does Shakespeare, as some have thought, intend in this play to preach [Col. 1c–2a]  any polemic against the life of the town, or compose any passionate encomium for the simple graces of the country. He is too great to be didactic on such narrow issues. Wherever men and women live, there they can be happy; or, as Touchstone says to Corin, when he expresses ignorance of the Court, one-sided enthusiasts for either streets or hedge-rows are truly damned, "like an ill-roasted egg, all on one side." Equanimity, such that of the Duke and old Adam, carries with it, wherever it be, the gracious gentleness of an eternal Eden.<ref>"St. James's Theatre." ''Daily Telegraph & Courier'' (London) 25 February 1890, Tuesday: 5 [of 12], Col. 6a–7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001112%2F18900225&page=5&article=048&stringtohighlight=mandeville+viscount. Print title: ''The Daily Telegraph'', p. 5. </ref></blockquote> The ''St. James's Gazette'' reported:<blockquote> We came intending to like it very much, and we succeeded in our intention. Indeed, we spent a really pleasant evening at the St. James's Theatre, Many of us knew one another to nod to, and some of us had bowing acquaintance with the author of the play; so it was all very sociable and comfortable. There were just enough shop-window celebrities to give an air of distinction to the rest of the audience, and everybody had a right to hope that his neighbours were wondering who he was. The critics were quite at their ease, having nothing to criticise except the acting; not being expected to suggest any slight alterations in the play (such as cutting out Acts I., III., and V., and rewriting Acts II. and IV.), but just to accept it as it stood, without even tracing it back to the French vaudeville from which it was, no doubt, originally adapted. The performance had many individual merits, but it also had two general faults. The first and least striking fault was that the interludes were  slurred over instead of being played up to. Now, the plot of "As You Like It" is so loose that a lengthy interlude (provided it be pretty) does not check the flow of interest; indeed, it gives a welcome rest and allows you to pick up your thoughts again. The other fault is one which may be, and is likely to be, remedied the play goes on. But at present it is neither smart nor spontaneous: the parts do not hang well together, and there is none of the enthusiasm which helps to pass off the interior performances of less skilful actors. The deficiency which we have noted was probably due to the serious illness from which Mrs. Langtry had been suffering and from which it was evident last night that she had not made a complete recovery. She worked very hard as Rosalind, and worked very well, and several times she made a really good hit; she was, perhaps, at her best in her indignant appeal to the usurper Duke against his unexpected and unjust decree of banishment. She was charming, again, in her embarrassment (as Miss Litton was charming before her) when she had her first meeting with her unconscious lover and she suggested very cleverly the absence of her skirts. She did her love-making archly and merrily; but though the pretended pretence gave her a right to go beyond the limits of maiden coyness, she seemed occasionally to assume the masculine Initiative when the feminine Suggestive would have been prettier. it was a pleasant though uneven performance, with plenty of grace in it and plenty of brains. The Touchstone of Mr. Sugden was in every way admirable — crisp, humorous, and distinctive. That his clowning was occasionally tedious was no fault of his. The palate of a modern audience does not relish these long-drawn pedantries of buffoonery. Our sense of humour is not more refined — it is different, and that's the end of it. But in the scenes with Audrey (and Miss Marion Lea made a capital Audrey — open-mouthed and innocently wicked or wickedly innocent) Mr. Sugden had a better opportunity and he made the most of it. There the humour was not conventional, and the audience enjoyed a laugh which was not scholarly. A word of praise must given to Mr. Brodie for his conscientious and effective rendering of the difficult and ungrateful part of the moonstruck lover of the wayward Phoebe. The first appearance of Mr. Bourchier in London attracted much interest. As an amateur at Oxford and elsewhere he was believed Iikely to to great things upon the stage, and his performance of Jaques was by no means a disappointment. The fault of it was that he did not suggest either the melancholy or the sententiousness which we associate with the character. The melancholy was but skin-deep, and the sententiousness simply was not there. But every actor has a right to read a part in his own way if it is a good way. And there is much to said for Mr. Bourchier's interpretation of Jaques. The temptation is to be stagy; Mr. Bourchier's tendency is to be natural. And he carried it out with great success in speaking the lines of the Seven Ages of Man. He caught up the Duke's phrase just as if it had dropped in the course of an ordinary conversation; and he played with it for a time until it seemed of itself to suggest to his mind the famous soliloquy. Certainly Mr. Bourchier’s method was effective, as it drew one of the heartiest rounds of applause in the whole evening. Among the company were noticed the following well known persons:— His  Royal Highness the Prince of Wales, the Duke and Duchess of Fife, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord De Grey, Mr. Alfred De Rothschild, Lord Lurgan, Sir Charles Russell, Sir Edward Guinness, Lady Colin Campbell, Lady Curzon, Lady Londonderry, Lady Hothfield, Hon. Alan Charteris, Mr. Oscar Wilde, Mr. Akers Douglas, M.P., Mr. Justin McCarthy, M.P., Sir Allen Young, Lord and Lady Coleridge, the Austrian Ambassador, the Spanish Ambassador, Lady Hozier, Mr. George Lewis, Mr. Whistler, Lady Greville [<nowiki/>[[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]]?].<ref>"As You Like It."  ''St James's Gazette'' 25 February 1890, Tuesday: 5 [of 16], Col. 1b–c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001485%2F18900225&page=5&article=027&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> This report is from ''Blalud, The Bath Society Paper'':<blockquote>Happily, there was no disappointment in store for the friends and admirers of Mrs. Langtry, who gathered on Monday night at the St. James’s Theatre in great force, to welcome this popular lady back to London and receive her with an extra warmth of cordiality in token of satisfaction at her recovery from her recent severe illness. Eight years have elapsed since she last played Rosalind in London. Since then she has travelled much, both at home and abroad, and followed the study and practice of her art with unfailing spirit and determination. She returns to the character which all leading actresses aspire to play in Shakespeare’s glorious dream of the Court of Duke Frederick and the forest of Arden a far riper artist, and, says the ''Daily News'', she presented us on Monday with an impersonation which compared to her former efforts is as the finished portrait to the crude sketch. Her performance may lack the great high tide of girlish impulsiveness, but first nights before an exceptionally critical audience and some remains of the languor of her recent indisposition doubtless contributed to rob her acting of something of its force and impulse. But her ''Rosalind'' remains a most winning and beautiful performance. A handsomer ''Rosalind'', or one that has worn the doublet and hose with more grace and picturesqueness (Mrs, Langtry, we may here note, scorns the modern heresy of boots, and appears in the more orthodox hose cross-gartered), has certainly not been seen in our time. The performance is full of good points; as when at the words, "here on the skirts of this forest like the fringe of a petticoat," she remembers with a momentary touch of maidenly shyness that petticoats no longer form part of her attire. Again, when in bantering her lover, who exhibits so strange an inability to recognize the voice and features of the woman whom he passionately loves, she pauses at "a beard neglected," and pointing at ''Orlando’s'' smooth chin bursts into uncontrollable laughter; again, when in the mock marriage scene, with ''Celia'' for the priest, her mind for the moment seems to wander away into such a dream as this mockery might suggest to one so deeply in love. The distinguishing feature of the impersonation, indeed, is the subtle suggestion of serious passion that underlies the frolicsome and bantering moods, and exhibits itself again and again in many subtle forms. The play is beautifully put on the stage, and if some of the performers fall a long way short of ideal excellence there are several impersonations that stand forth in strong relief and are of far more than common excellence. The ''Phœbe'' and ''Silvius'' of Miss Beatrice Lamb and Mr. Matthew Brodie deserve special note: but the most brilliant success of all in the case of the minor characters was undoubtedly the ''Audrey'' of Miss Marion Lee, whose hoydenish, but not ungraceful antics, whose happy, contented ignorance, whose mingled wonderment and admiration at ''Touchstone's'' philosophical utterances were admirable. Everything that taste can suggest has been done to embellish the play; the musical features are strengthened by an efficient chorus, and the costumes designed by the Hon. Lewis Wingfield are rich and handsome. Perhaps it might be objected that Mrs. Langtry’s beautiful green velvet doublet, with its slashes, tags, and laces, her cinnamon jerkin, her black cap and jewelled buckle, and her rich and massive cloak were hardly suited to her daring enterprise on the highways and in the forests of the Duke’s dominions. The restoration of Hymen, gracefully represented by Miss Violet Armbruster, and the final morris dance in honour of the nuptials, gave the crowning fanciful touch to the performance, and prepared the spectators for the playful epilogue, which was spoken by Mrs. Langtry, in her simple nuptial white robe and leopard skin, with rare elocutionary grace and effect,. [sic] The audience which greeted Mrs. Langtry was at once distinguished and interesting. The Royal box was occupied by the Prince and Princess of Wales, the Duchess of Fife, the Princesses Victoria and Maud of Wales, and the Duke of Fife. Among the representatives of "Society" were Lady Londonderry, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord de Grey, Lord and Lady Coleridge, Lady Curzon, Lady Hothfield, Lord Lurgan, Lady Greville [<nowiki/>[[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]]?], Lady Hozier, Mr. Alfred de Rothschild, Sir Charles Russell, Sir Edward Guinness, Mr. Coningsby Disraeli, Mr. Akers Douglas, M.P., Mr. Justin M‘Carthy, M.P., Sir Allen Young, Lady Colin Campbell, Lady Monckton, Mrs. Labouchere, and the Austrian and Spanish Ambassadors. Pictorial art was represented by Mr. Whistler; music by Mdme. Antoinette Sterling; dramatic writing by Mr. H. A. Jones; and theatrical interests by Mrs. Bancroft, Miss Genevieve Ward, Mr. and Mrs. John Lart, Mrs. John Hare, Mr. M. L. Mayer, Mr. Augustus Harris, &c.<ref>"Mrs. Langtry." ''Bladud'' 26 February 1890, Wednesday: 9 [of 16], Col. 2b–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005771%2F18900226&page=9&article=055&stringtohighlight=mandeville+viscount. Print title: ''Blalud, The Bath Society Paper'', p. 9.</ref></blockquote> === 25 February 1890, Tuesday === On 5 March, the ''World'' reports on a party hosted by Mrs. Adair that must have been held the week before:<blockquote>Many smart people were present at Mrs. Adair's party in Curzon Street on Tuesday night. The rooms, lighted by electricity, looked their prettiest with floral decorations consisting entirely of tulips, the hostess appearing in a gown of terra-cotta velvet trimmed with dark fur. The Austrian Ambassador and Madame Waddington were among those who listened to Miss Damian’s rendering of a new song by Tosti, and to the glees sung by Mr. Avon Saxon's American quartet; and among others present were Lady Charles Beresford, [[Social Victorians/People/Mandeville|Lady Mandeville]], the Duchess of Marlborough, Lady Fanny Marjoribanks, who brought her sister, Lady Sarah Churchill, Sir Arthur Sullivan (just back from his annual sojourn at Monte Carlo), Lord and Lady Wharncliffe, Lady Howe and Lady Edith Curzon, Lord and Lady William Nevill, Mrs. Cavendish-Bentinck, General Charles Fraser, Lady Conyers, Madame de Brienen, and Sir Philip Currie.<ref>"What the World Says." ''The World'' 5 March 1890, Wednesday: 19 [of 44], Col. 2a [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0006339%2F18900305&page=19&article=061&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote>The ''Daily Telegraph'' covered the party with a few additional names:<blockquote>A casual passer-by in Curzon-street last night might almost have imagined from the long string of carriages assembled there that, in spite of the cold east wind blowing, the London season had already begun. Mrs. Adair had together to hear some beautiful music, including a celebrated quartette of American singers, a large and fashionable crowd, composed of the ''élite'' of London society, comprising among many others, the Austrian Ambassador, the French Ambassadress, the Marquis and Marchioness of Tweeddale, Lady Charles Beresford, Lady Fanny Marjoribanks and Lady Sarah Spencer-Churchill, Earl and Countess of Wharncliffe, Countess Howe and Lady Edith Curzon, Lord and Lady William Nevill, Lord George Nevill, Viscount and Viscountess Curzon, Countess of Sefton and Lady Rose Molyneux, Viscountess Mandeville, Julia Countess of Jersey and Mr Brandling, Viscount and Viscountess Bury and the Hon. Mary Keppel, the Countess of Crawford, the Earl of Clonmell, the Countess of Antrim and Lady Louisa Beauclerk, Viscountess Melgund, Lady Conyers, Hon. Lady Cotterell and Miss Cotterell, Lord Alington, Lord Alcester, Lord Greenock, Viscount de Vesci, Lady Hilda Brodrick, Hon. W. Lowther, M.P., and Mrs. Lowther, Hon. Mrs. Bampfylde, Hon. Hallyburton Campbell, Mrs. and Miss Campbell, Hon. Kenneth Howard, Hon. Derek Keppel, Hon. FitzRoy and Mrs. Stewart, Lady FitzGerald, Mrs. Bischoffsheim, Madame de Brienen, Lieutenant-General Sir Seymour Blane, General C. Fraser, M.P., Mr. Chamberlain, M.P., Mr. Howard Vincent, M.P., and Mrs Vincent, Mr. Charles Hall, M.P., Mrs. Cavendish-Bentinck, Sir C. Rivers Wilson, Sir Arthur Sullivan, Sir Philp Currie.<ref>"London Day by Day." ''Daily Telegraph & Courier'' (London) 26 February 1890, Wednesday: 5 [of 8], Col. 4a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001112%2F18900226&page=5&article=094&stringtohighlight=mandeville+viscount. Print title: ''The Daily Telegraph'', p. 5. </ref></blockquote>The ''Sporting Gazette'' also had a report, on 1 March:<blockquote>Mrs. Adair's party in Curzon-street on Tuesday night was, as usual, exceedingly smart and dressy. The "smarties" turned out strong for this Anglo-American function, and wore their best frocks and jewels. Lady Tweeddale, Lady Howe, with Lady Edith Curzon, Lady Fanny Majoribanks chaperoning her younger sister, Lady Sarah Churchill, Lady Curzon, Lady Antrim with Lady Louisa Beauclerk, Lady William Nevill with some lovely diamonds, [[Social Victorians/People/Mandeville|Lady Mandeville]], Lord Alington, Sir Arthur Sullivan, Mrs Cavendish Bentinck, and others. The singing of an American quartette was ''the'' feature of the evening.<ref>"The Man About Town." "The County Gentleman." ''Sporting Gazette'' 1 March 1890, Saturday: 6 [of 38], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0002525%2F18900301&page=6&article=053&stringtohighlight=mandeville+viscount. Same print title, p. 274.</ref></blockquote> ==March 1890== === 10 March 1890, Monday === The Prince and Princess of Wales hosted a ball in celebration of their wedding anniversary. This report from the ''Bedfordshire Mercury'' is very brief:<blockquote> The Prince and Princess of Wales, in celebration of the anniversary of their wedding on March 10, 1863, gave a dance at Marlborough House on Monday night. The Danish Minister and Madame de Falbe were among the company invited, also the Earl and Countess Brownlow, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lord Alwyne Compton, Baron F. de Rothschild, &c.<ref>"Notes of the Aristocracy." ''Bedfordshire Mercury'' 15 March 1890, Saturday: 7 [of 8], Col. 4b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001290%2F18900315&page=7&article=073&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> The report from the ''Morning Post'' suggests that this was a smaller party than usual for the Prince and Princess of Wales:<blockquote>In celebration of the anniversary of the Prince and Princess of Wales's wedding day, their Royal Highnesses gave an evening party (dancing) at Marlborough House last night. His Royal Highness the Duke of Edinburgh, her Royal Highness Princess Beatrice (Princess Henry of Battenberg) and his Royal Highness Prince Henry of Battenberg, her Royal Highness the Princess Louise of Wales (Duchess of Fife) and the Duke of Fife, his Royal Highness Prince George of Wales, her Royal Highness the Princess Victoria of Wales, her Royal Highness the Princess Maud of Wales, her Royal Highness the Princess Mary Adelaide (Duchess of Teck), and his Highness the Duke of Teck and her Serene Highness the Princess Victoria of Teck and his Serene Highness Prince Francis of Teck, and their Serene Highnesses the Prince and Princess Victor of  Hohenlohe-Langenburg and the Countess Victoria Gleichen were present. The following had the honour to receive invitations, but some amongst them were unavoidably prevented from attending: — The French Ambassador (M. Waddington) and Madame Waddington, the German Ambassador (Count Hatzfeldt) and the Countess Marie Hatzfeldt, the Turkish Ambassador (Rustem Pasha), the Austro-Hungarian Ambassador (Count Deym) and the Countess Deym, the Italian Ambassador  (Count Tornielli), the Countess Tornielli, and Mademoiselle Virginia dei Conti Lazzari, the Danish Minister (M. de Falbe) and Madame de Falbe, the Russian Chargé d'Affaires (M. Bontineff), his Serene Highness Prince Louis Esterhazy, M. A. Kroupensky, M. A. de Stoeckl, Count Leyden, Count Albert Mensdorff, Count Reventlow, Mr. and Mrs. Henry White, M. de Soveral, the Marques and Marquesa de Santurce, Mr. A. de Murrieta, Mr. C. de Murrieta. The Duke and Duchess of Leeds and Lady Alice Osborne, the Duke and Duchess of Buccleuch, the Duke and Duchess of Portland, the Duke and Duchess of Westminster, the Marchioness of Blandford and Lady Frances Spencer Churchill, the Marquis and Marchioness of Granby, the Marquis and Marchioness of Londonderry, the Marquis and Marchioness of Ormonde, the Marquis and Marchioness of Salisbury and Lady Gwendolen Cecil, the Marquis and Marchioness of Stafford, Count Gleichen, the Earl and Countess Brownlow, the Earl and Countess Cadogan and Lady Emily  Cadogan, the Earl and Countess of Clarendon, the Earl of Chesterfield, the Earl of Clonmell, the Earl of Dudley, the Earl and Countess of Dunraven and Lady Florence Wyndham Quin, the Earl and Countess Granville and Lady Victoria Leveson Gower, the Earl and Countess of Hardwicke, the Earl and Countess Howe and Lady Evelyn Curzon, the Earl and Countess of  Kilmorey, the Earl and Countess of Lathom and Lady Alice Wilbraham, the Earl and Countess of Lucan and Lady Rosalind Bingham, the Earl of Mount Edgcumbe and Lady Albertha Edgcumbe, the Earl and Countess of Romney and Lady Florence Marsham, the Earl and Countess of Rosebery, the Earl of Shaftesbury, the Earl of Stradbroke, the Earl and Countess of Sefton and Lady Gertrude Molyneux, the Earl and Countess Spencer, Viscount Drumlanrig, Lord and Lady Randolph Spencer Churchill, Lord Frederick Hamilton, Lord Ernest Hamilton, Lord and Lady Algernon Gordon Lennox, [[Social Victorians/People/Mandeville|Viscountess Mandeville]], Lord Charles Montagu, Viscount Chelsea, Viscount and Viscountess Coke and the Hon. Violet White, Viscount Curzon, Lady Georgina Curzon and Lady Sarah Spencer Churchill, Viscount Fincastle, Viscount Royston, Viscount Valletort, Lord and Lady Brooke, Lord Burghersh, Lord Capell, Lord and Lady Norreys and the Hon. Coralie Glyn, Lady Charles Beresford, Lord Marcus Beresford, Lord Alwyne Compton, Lord and Lady William Nevill, Lord Richard Nevill, Lord Henry Paulet, Lord Henry Vane Tempest, Lord Herbert Vane Tempest, Admiral Lord Alcester, Lady Ampthill and the Hon. Constance Russell, Lord Alington and the Hon. Winifred and Hon. Mildred Sturt, Lord and Lady Colville of Culross  and the Hon. Blanche Colville, Lord and Lady De  Ramsey, Lady Hastings, Lord and Lady Hindlip, Lord Lurgan, Lord and Lady Revelstoke and the Hon. Susan Baring, Lord and Lady Rothschild,  Lord Rowton, Lord Suffield and the Hon. Judith and Hon. Eleanor Harbord, Lord and Lady Wimborne and the Hon. Frances Guest, Lord Wolverton, Baron Ferdinand Rothschild, Captain the Hon. Maurice Bourke, R.N., Colonel the Hon. Henry Byng and the Misses Byng, the Hon. Alan Charteris, Lieutenant-Colonel the Hon. C. Edgcumbe, the Hon. Sir Spencer and Lady Ponsonby Fane, the Hon. Sidney Greville, Colonel the Hon. Oliver Montagu, the Hon. C. R. and Mrs. Spencer, the Hon. S. M'Donnell, the Hon. Alexander Yorke, the Hon. H. and Lady Feodora Sturt, the Hon. Arthur Walsh, the Hon. R. Winn, the Right Hon. W. E. and Mrs. Gladstone and Miss Gladstone, the Right Hon. A. J. Balfour, the Right Hon. H. Chaplin, the Right. Hon. Sir Henry James, the Right Hon. Sir C. L. Wyke, the Hon. S. C. Colville, the Hon. Assheton Harbord, the Hon. Julia Stonor, the Hon. H. Stonor, the Hon. E. Stonor, Captain the Hon. Henry White, Sir Edward and Lady Guinness, Sir Charles Hartopp, Sir Frederick Johnstone, Sir John and Lady Lister Kaye, Sir James and Lady  Mackenzie, Colonel Sir Nigel and Lady Emily Kingscote, Admiral Sir R. and the Hon. Lady Macdonald and Miss Macdonald, General Sir Dighton and Lady Probyn, Sir Francis and the Hon. Lady Knollys, Major-General Sir Christopher Teesdale, Sir Oscar Clayton, Sir Allen Young, Lieutenant-General Bateson, Lieutenant-General Charles Fraser, Major-General and the Hon. Mrs. Arthur Ellis and Miss Ellis, Colonel J. P. Brabazon, Colonel and Mrs. Stanley Clarke and the Misses Clarke, Captain H. F. Stephenson, R.N., Lieutenant-Colonel F. H. Poore, Lieutenant-Colonel R. Vivian, Major C. Swaine, Captain Douglas Dawson, Captain C. H. A. Hervey, the officers of the Queen's Guard, St. James's Palace, Mr. G. Byng, Mr. Henry Calcraft, Mr. Victor Cavendish, Mr. Henry Ellis, Mr. Horace Farquhar, Mr. and Mrs. George Forbes and Miss Forbes, Mr. Montagu Guest, Mr. Charles Hall, Mr. E. W. Hamilton, Mr. and Lady Hilda Higgins, Mr. Holzmann, Miss Knollys, Mr. and Mrs. F. R. Knollys, Mr. Ronald Moncrieffe, Mr. Alfred Montgomery, Mr. Arnold Morley, Mr. Henry Petre, Mr. Alfred de Rothschild, Mr. and Mrs. Leopold de Rothschild, Mr. Reuben Sassoon and Miss Sassoon, Mr. and Mrs. Arthur Sassoon, Mr. Edgar Sebright, Mr. and Mrs. Ralph Sneyd, Mr. Christopher Sykes, and Mr. O. G. St. George Williams. Messrs. Coote and Tinney's band was in attendance.<ref>"Marlborough House." ''Morning Post'' 11 March 1890, Tuesday: 5 [of 8]. Col. 6b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000174%2F18900311&page=5&article=055&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> ==April 1890== Arnold Dolmetsch took out a three-day ticket to the Reading Room in April 1890 (Campbell 22). === 9 April 1890, Wednesday === ==== The New Forest United Hunt Ball ==== <blockquote>This annual ball was beld at the New Forest Hall, Lyndhearst, on Wednesday night, and was a most brilliant gathering, upwards of 200 being present, not only the New Forest but also neighbouring hunts being well represented. Among those present were Earl and Countess of Londesborough, Lady Mildred Denison, Lady Lilian Denison, Hon. Louis Johnstone, Hon. Gerald and Mrs Lascelles, Major Downman, Capt. Sandbach, R.H.A., Col. and Mrs Sinkins, Col. Fitzroy Smyth, Lady Thursby and party, Mr Stanley Pearce (Master of the Foxhounds), Mr F. Lovell (Master of the Deerhounds) and party, Lady Adela Goff and party, Hon. Miss St. Leger Glyn and party, Mr A. H. Jeffreys, M.P., and Mrs Jeffreys, Hon. Mrs A. Peter, Mr Somerset H. Onslow, Mr Lancelot Orde, Lord Garioch, Mr and Mrs C. K. Francis, the Countess of Normandy, Lady Mary Agar, Lord Montagu, Hon. John Scott Montagu, Viscount Cantelupe, Prince Dhuleep Singh, Hon. D. Keppel, Col. Charles Needham (1st Life Guards), [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], Mrs Cornwallis West, Mr and Mrs Shelley Bontein, Col. Gridley, Col. Newton, Col. Elliot, R.H.A., Lady Meyrick and party, Sir W. Jervois, Lady Jervois and party, &c.<ref>"The New Forest United Hunt Ball." ''The Sportsman'' 11 April 1890, Friday: 2 [of 4], Col. 5c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001055/18900411/024/0002. Same print title and p.</ref></blockquote> ==May 1890== Leonora Braham appeared at the Opera Comique as Ruth Ferns in ''Gretna Green''. ===21 May 1890, Wednesday=== It was his pupil the young Countess Valda Gleichen who found a sponsor for Dolmetsch's first but brief appearance at a London concert on 21 May 1890, when the 'Magpie Minstrels' … gave one of their bi-annual performances at the Princes' Hall" (Campbell 30). ==June 1890== === 3 June 1890, Tuesday === ==== Dinner and Concert Hosted by Mrs. Arthur Williams and Ball by Mrs. Menzies ==== <blockquote>Yesterday evening society passed a large proportion of time on wheels, and the number of receptions was more than the most inveterate could get through wvith comfort. ... Meanwhile, the Duke of Clarence and Avondale dined with Mrs. Arthur Wilson in Grosvenor-place, and a small and select concert followed. The Duchess of Leinster was, as ever, the most beautiful among many beauties, the Hon. Helen Henniker, after her long mourning, was warmly greeted by swarms of friends, and the Countess of Lovelace, in grey satin, chaperoned her débutante niece, Miss Milbank. Prince "Eddie" went from Mrs. Arthur Wilson's hospitable mansion to that of Mrs. J. Graham Menzies. It is unusual to find mother and daughter entertaining this illustrious young man on the same evening. Mrs. Menzies, who has succeeded Lord Suffield in his beautiful house in Upper Grosvenor-street, had one of the smartest and most successful balls of the present season. The Blue Hungarians played with immense vigour and owing to the postponement of Lady Carnarvon's ball, there was no competition for men and married women. Countess Howe, Lady Brougham and Vaux, Lady Aveland and Miss Willoughby, Viscountess Stormont, the Hon. Lady Grey Egerton and Miss Egerton, the Duchess of Buccleuch, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Earl of Craven, Countess of Coventry, Mrs. Mackay, Lady Sykes, the Marchioness of Granby, Mrs. Hwfa Williams, Lord Garioch, Sir Savile and Lady Crossley, Mr. and Mrs. Harry Lawson, and many others were present.<ref>"Yesterday Evening Society." ''Pall Mall Gazette'' 4 June 1890, Wednesday: 4 [of 8], Col. 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18900604/015/0004. Same print title and p.</ref></blockquote> ==== Münster-Hay Wedding ==== Account of the wedding of Count Alexander Münster and Lady Muriel Henrietta Constance Hay. Count Bismarck sent a gift.<blockquote>St. Andrew's Church, Wells-street, was completely filled yesterday afternoon on the occasion of the marriage of Count Alexander Münster, son of Count Münster, German Ambassador in Paris, with Lady Muriel Henrietta Constance Hay, younger daughter of the Earl and Countess of Kinnoull. The ceremony was fixed for half-past two o'clock, but all the seats in the body of the church were occupied long before that time by invited guests. The rood screen was hidden with white flowers, and a number of tall palms were skilfully arranged in and about the chancel. The bridegroom was early in his place, and wore the splendid white and gold uniform of the Garde du Corps of the Emperor; his best man, Prince Hans Heinrich Pless, wearing the scarlet and gold uniform of the Garde Hussars. The six bridesmaids were Lady Mildred Denison, Lady Dorothea Stewart Murray, Hon. Marie Hay, Hon. Marjory Murray, Miss Rosalind Lovell, and Miss Gladys Hadow (niece of the bride). They wore pretty dresses of pale blue satin, veiled with white chiffon; the skirts were edged with a frill, and the bodices finished with quaint fichus of the chiffon, fastened in front with a bunch of forget-me-nots. They wore drawn hats of the chiffon, covered with forget-me-nots, and their posies were of the same flowers, intermixed with pink roses, and tied with narrow pink ribbons. Each wore a gold bangle with centre forming the initials "A.M.," and a single turquoise drop. The bride entered the church at half-past two, and was received by her father, who led her to the chancel steps, preceded by the choristers singing a nuptial hymn. The bride's dress was of ivory satin made with a very long train. The front was fringed round the hem with orange blossoms and veiled with old Brussels lace, the gift of her mother; the bodice was entirely covered with beautiful old lace, and finished at the neck on one side with a cluster of orange blossoms. She wore a small wreath of orange blossoms and myrtle arranged in the hair at the back and a tulle veil falling almost to the end of the train and held by diamond stars, the gift of the bridegroom. The bride's other jewels included two diamond drop hairpins, the gift of Count Münster; an emerald and diamond brooch, from the bridegroom; enamel and diamond heartsease brooch, from Lady Mary Mills; a diamond and emerald bracelet, from the Duke of Beaufort; and a diamond bracelet, from the Duke and Duchess of Athole. Her train was held by her two little nephews, Masters Roland and Patrick Hadow, wearing Scotch dress with lace ruffles. The service, which was fully choral, was performed by the Rev. Provost Rorison, of St. Ninian's Cathedral, Perth, assisted by the Rev. W. T. Houldsworth, rector of St. Andrew's. The Earl and Countess of Kinnoull afterwards entertained the bridal party at the residence of the Earl of Londesborough, uncle of the bride, in Grosvenor-square. Among the numerous guests were the Russian Ambassador and Madame de Staal, the German Ambassador, the Austrian Ambassador and Countess Deym, Count Münster and the Countesses Marie, Julia, and T. Münster, Prince Pless, Prince Louis Esterhazy, Count Donhoff, Baroness von Beckendorff-Hindenburg, Baron E. Fürstenberg, Count Paul Metternich, Geheimrath Schmettau, Countess Valentin, [[Social Victorians/People/de Soveral|M. de]] [[Social Victorians/People/de Soveral|Soveral]], Captain Hasenclever, Count Leyden, Herr von Rath, Count Francis Deym, Baron and Baroness Deichmann, Count Mensdorff, Herr Pilgrim Baltazzi, Count and Countess Henry Lützow, Baroness d' Erlanger [?], Herr von Kemnitz, Baron Cramm, Vicomte de Janzé, the Duke of Beaufort, the Duchess of Athole, and the Ladies Stewart Murray, the Marchioness of Downshire, the Marchioness of Stafford and Lady Sybil St. Clair Erskine, Maria Marchioness of Ailesbury, the Countess of Dunraven and Ladies Wyndham Quin, the Countess of Kilmorey, the Countess of Galloway and Miss Stewart, the Countess of Dudley and Lady Edith Ward, the Countess of Strathmore and Lady Constance Lyon, the Earl and Countess of Roden, Lady Willoughby de Eresby and Hon. Miss Willoughby, the Countess of Westmorland. Viscount and Viscountess Raincliffe, Viscountess Coke, Viscountess Stormont and Hon. MabeI Murray, Viscountess Malden and Hon. Miss Capel, Lord Hay of Kinfauns [?], Lady Delamere and Miss Cholmondeley, Lady Brougham, Lady Esher, Lord Saye and Sele and Hon. Maud and Hon. Gertrude Fiennes, Lady Helen MacGregor and Miss MacGregor, Lord and Lady Algernon Gordon Lennox and Lady Angela St. Clair Erskine, Lady Raglan, Lord Charles Innes Ker, Lady Elizabeth Bertie, Lady Frances Pratt and Lady Clementine Pratt, Lady Geraldine Somerset, Lady Dalton Fitzgerald and Miss Payne, Lady Mary Mills and Miss Pereira, Lady Dorchester, Lady Louisa Moncreiffe and Misses Moncreiffe, Lady Norreys, Lady Petre, Lady Jane Trefusis, Lord Frederic Hamilton, Baroness de Brienen and Miss de Brienen, Helen Lady Forbes and the Misses Forbes, Lady Scott, Miss Scott and Mrs. Green, Lord Zouche and Hon. Miss Curzon, Mr. and Lady Constance Hadow, Hon. Lady Campbell, Lady William Nevill and Miss Bannerman, Hon. Alastair and Mrs. Hay, Hon. Miss Greville, Captain Hon. Randolph Stewart, Hon. FitzRoy and Mrs. Stewart, Hon. Charles Walsh, Hon. William Walsh, [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], Colonel Hon. Charles and Mrs. Hay and Miss Maude Hay, Hon. Mrs. Greville Nugent, Hon. Mrs. Colville and Miss Streatfeild [sic], Hon. Claude Hay, Sir Charles and Lady Forbes of Newe, Laely Borthwick, Sir Francis Montefiore, Sir Henry Hoare, Sir George Arthur, Sir William Gordon Cumming, General and Mrs. Wilkinson, Colonel and Mrs. Chaine, Colonel Arthur Paget, Colonel H. M. Hozier, the Provost of St. Ninian's, Major Finch, Major Candy, Colonel Lloyd, Captain and Mrs. Stewart, Mr. and Mrs. Hay Drummond, Captain and Mrs. Lloyd, Mr. William Gillett, Mr. Hulse, M.P., and Mrs. Hulse, Captain Lumley, Captain Barrow, Major Anstruther, Mrs. Hwfa Williams, Mr. Arthur Hay, Mr. Raglan Somerset, Mr. and Mrs. George Cavendish Bentinck, Mrs. Villiers, Mr. Henry Petre, Mrs. Keighley, Mrs. E. Balfour and Miss Balfour and Miss Ellice, Mrs. Dick Cunynghame and Miss A. M'Gregor, Rev. C. Weidemann, Miss Balfour and Miss E. Balfour, Mrs. and Miss Halford, Mr. and Mrs. Toler, Mrs. and Miss Townley, Rev. Dr. M'Ewan, Mr. A. F. Levita, Mr. F. Stephens, Mrs. and Miss Leigh, Mr. and Mrs. George Forbes and Miss Forbes, Mrs. Luke Wheeler, Mrs. J. Beerbohm, Dr. and Mrs. W. M. Collins, Mrs. and Miss Orde, Mrs. Murray, Miss Glyn, Mrs. Godfrey Pearse, Mr. Balfour, Mr. Leopold Goldberg, Mr. and Miss Oswa d [Oswald], Mr. and Mrs. C. Bulteel, Mrs. Brydges Wyllyams, Mr. H. Balfour, the Misses Codrington, Miss Finch, Mrs. Barron and Miss Sullivan, Mr. Hamilton Aidé, Mr. Granville and Miss Berkeley, Mr. M'Gregor, Mrs. and Miss Needham, Mr. and Mrs. A. Consett and Miss Consett, Mrs. Herbert Naylor LeyIand, Mr. and Mrs. A. Darby, Mrs. Montefiore, Mrs. Charles Murray and Miss Murray, Mrs. Thomas Legh, the Rev. John F. Mitchell, Mr. and Mrs. Claud Hay, Mr. Fiztalan Manners, Mr. Albert Stopford, Mr. Hervey Pechell, and Mr. and Mrs. Charles Orde. Later in the day Count Alexander Münster and his bride left for Hillingdon Court, Uxbridge, the residence of the Hon. A. Mills, where they will pass the early days of the honeymoon, and then proceed to Chàlet Sans Souci, Bellaggio, Lake of Como, lent by Mr. Hervey Pechell. They will afterwards go to Wiesbaden, and thence to Count Münster's German country seat, where a great reception awaits them. The wedding presents were very numerous. Count Munster presented the bride with a diamond necklace and pair of diamond drop hairpins. The bridegroom's gifts to his bride included a tiara of three diamond stars, a brooch formed of an emerald of unusual size set in diamonds, large pearl pendant, gold bracelet set with turquoises, and diamond and turquoise ring. The Duke of Beaufort presented Lady Muriel with an emerald and diamond bracelet. The Earl and Countess of Kinnoull's presents included a quantity of plate and other articles. Prince Henry of Battenberg sent a large silver-mounted claret jug; Count Bismarck, a silver box; the Marchioness of Waterford, gold snake bracelet, with pearl head and ruby eyes; the Duke and Duchess of Athole, diamond "M" bangle; Lady Ermyntrude Malet, massive silver bowl; Countess Marie Münster, silver tea set and tray; Baron E. Fürstenberg, pair of gold-mounted claret jugs; Baron and Baroness Erlanger, large silver bowl; Baron and Baroness Gungberg, pair of antique silver mounted jugs.<ref>"Marriage of Count Alexander Münster and Lady Muriel Hay." ''Morning Post'' 4 June 1890, Wednesday: 7 [of 12], Col. 7a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900604/064/0007. Same print title and p.</ref></blockquote> ===4 June 1890, Wednesday=== Derby Day. According to the Morning Post, <quote>Lady Thurlow's first dance. / Mrs. Coddington's Ball. / The Fourth of June celebration at Eton. / ... Royal Society of Literature — Paper on "Early England," by Mr. Carmichael, S. </quote> ("Arrangements for This Day." The Morning Post Wednesday, 4 June 1890: p. 7 [of 12], Col. 6A). ===13 June 1890, Friday=== ? or on the 12th? Arnold Dolmetsch had a concert at which his students performed (Campbell 31). It was reviewed very positively in the 13 June Daily Graphic, especially Helene Dolmetsch's performance. They did Purcell's Fairy Queen. They used a harpsichord. ===26 June 1890, Thursday=== There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, [[Social Victorians/People/Arthur Sullivan|Arthur Sullivan]], Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902. ==July 1890== ===3 July 1890, Thursday=== <blockquote>On July 3, 1890, Besant’s house at 19 Avenue Road, St. John’s Wood, was inaugurated as a new center of Theosophical work, serving as the European headquarters of the [[Social Victorians/Theosophical Society|Theosophical Society]] and as HPB’s residence. That house became also the meeting place of HPB’s Inner Group, twelve of her students who were eager for more intense training than was feasible in a larger, more general gathering.</blockquote>That Inner Group is different from the Esoteric Section.<ref>"Esoteric World Chapter 23." ''The Theosophical Society of America, Online Resources''. http://www.theosophical.org/index.php?option=com_content&view=article&catid=23&id=1725&Itemid=53</ref><blockquote>The hall had a seating capacity of about 200. On the opening night [July 3, 1890] [sic] the room was crammed, and many were unable to gain admission. The speakers were Mrs. Besant, Mr. Sinnett, a Mrs. Woolff (of America), and Mr. [Bertram] Keightley. HPB was present but said nothing, on account of the critical state of her health.<ref>"23a. Henry S. Olcott, July 1890, London [Olcott 1931, 4:254–6]." Esoteric World Chapter 23. ''The Theosophical Society of America, Online Resources''. http://www.theosophical.org/index.php?option=com_content&view=article&catid=23&id=1725&Itemid=53.</ref></blockquote> === 4 July 1890, Friday, 11 p.m. === ==== The Queen's State Ball at Buckingham Palace ==== The Prince and Princess of Wales were hosts of this ball, as Queen Victoria was not present. Many of the Queen's children and their spouses were present.<ref>"State Ball at Buckingham Palace." ''Morning Post'' 5 July 1890, Saturday: 5 [of 8], Col. 5a–7b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900705/038/0005. Same print title and p.</ref> The dancing commenced shortly after the Prince and Princess of Wales arrived at 11:00. === 1890 July 15, Tuesday === ==== Garden Party at Marlborough House to Meet the Queen ==== The beginning of the ''Morning Post'' article, which is followed by a very long list of people invited:<blockquote>Their Royal Highnesses the Prince and Princess of Wales gave a garden party at Marlborough House yesterday afternoon to meet the Queen. Her Majesty was accompanied by their Royal Highnesses the Princesses Victoria and Margaret of Prussia, and was attended by the Duchess of Roxburghe, Miss M'Neill, the Hon. Evelyn Moore, the Hon. Bertha Lambert, and Major-General Sir Henry Ewart and Major F. A. Bigge. Madlle. De Perpigna was in attendance on the Princesses Victoria and Margaret. The following were present: — Their Royal Highnesses the Duke and Duchess of Edinburgh and Prince Alfred, and the Princesses Marie, Victoria, Alexandra, and Beatrice of Edinburgh; their Royal Highnesses the Duke and Duchess of Connaught and Prince Arthur, and the Princesses Margaret and Patricia of Connaught; their Royal Highnesses the Prince and Princess Christian of Schleswig-Holstein Sonderburg-Augustenburg and Princess Victoria and Princess Louisa of Schleswig-Holstein, her Royal Highness the Princess Louise, Marchioness of Lorne, and the Marquis of Lorne. Her Royal Highness the Duchess of Albany, her Royal Highness the Princess Frederica and the Baron von Pawell-Rammingen, his Royal Highness the Duke of Cambridge, her Royal Highness Princess Mary Adelaide Duchess of Teck and his Highness the Duke of Teck and Princess Victoria of Teck, his Royal Highness the Duke d'Orleans and Princess Hélène d'Orleans, his Imperial Highness Don Pedro Augustus of Coburg and Braganza, his Serene Highness the Prince of Schaumburg-Lippe, his Serene Highness the Prince of Leiningen and their Serene Highnesses Prince and Princess Victor of Hohenlohe-Langenburg and the Countesses Gleichen, and Count Gleichen.<ref>"Garden Party at Marlborough House." ''Morning Post'' 15 July 1890, Tuesday: 5 [of 8], Col. 3a–7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900715/044/0005. Same print title and p.</ref></blockquote> === 19 July 1890, Saturday === ==== Wedding of James Francis Harry St. Clair-Erskine and Violet Aline Vyner ==== This report from the ''Fife Free Press'' is missing a big piece of the 2nd paragraph, where the bridesmaids' and bride's dresses are described, because the page was damaged. The digitized copy is also generally poor.<blockquote>On Saturday forenoon, in Michael’s Church, Chester Square, London, was celebrated the marriage of Lord Loughhorough, eldest son of the Earl of Rosslyn with Miss Violet Aline Vyner, younger daughter of Mr R. C. Vyner, of Fairfield, Yorkshire. The chancel was beautifully decorated with palms and a profusion of lilies most artistically arranged. The Prince of Wales, Duke of Cambridge. Princess Mary Adelaide, Duchess and the Duke of Teck, and Princess Victoria, and the Right Hon. W. E. Gladstone were among those present. The wedding party began to arrive at 11 o’clock, the bridesmaids being early in attendance. They were the Ladies Sybil and Angela St Clair Erskine, sisters of the bridegroom; Lady Anne Lambton, Miss Musgrave, Miss Rachel Gurney, and Miss Fleming. They were charmingly dressed in soft cream crepon, with violet velvet Zoave jackets, forming half sleeves and capes at the back, and pretty turn-down collars and cravats of white mosseline [mousseline?] chiffon, and the skirts were very plainly made. They wore large white hats lined with a band of violet velvet near the face, and trimmed on the crown with white ostrich plumes, a bunch of violets fastening the brim at the back. Each wore a brooch — the St Clair cross in rubies and diamonds set in a circle of the latter, and surmounted by a baron’s cornet of the same precious stones — the gift of the bri[degroom]; and carried a bouquet of Marechal Niel [roses?] ...ging sprays of the same, and tied w... ribbons. The bride entered the ... eleven o’clock, being received at ... father, who led her to the cha... her away. The service be... procession was preceded up ... choir, who sang "The voic..." The bride’s dress was ... with a long court train, ... white velvet being embro... ...und the train, and fasten... ...ver true lover’s knot. The picture... ...gh silver waistband, had full slee... mousseline chiffon; and a small ... blossoms was most becomingly arra... with a long tulle veil. The Earl of Chesterfield discharged the duties of best man. The Rev. Canon Fleming, Chaplain in Ordinary to the Queen and Vicar of St Michael's, performed the marriage rite, assisted by the Rev. Edgar Sheppard, Sub-Dean of the Chapels Royal, and the Rev. J. Thomson, of Rosslyn, near Edinburgh. The hymn “Thine for ever! For of love,” was sung the end of the service. Canon Fleming afterwards delivered a most impressive address. After the benediction the clergy and bride and bridegroom passed to the vestry, the Prince of Wales following with Mrs Vyner. Pretty wedding favours of orange blossoms, Parma violets, and white heather were then distributed amongst the congregation. Mr and Mrs Vyner afterwards welcomed the wedding party at Ashburnham House, Dover-street. The ''dejeuner'' was served in three rooms, one being reserved for the bride and bridegroom and the Royal party. The table in the latter room was decorated with several of Mr Vynet's racing trophies. The band of the Grenadier Guards performed in the garden during the reception. The Prince of Wales, Duke of Cambridge, the Duke and Duchess of Teck and Princess Victoria were present at the breakfast, and among the general company were Count Munster and Countess Maria Munster, the Marchese and Marchesa de Montagliari, the Marquis and Marchioness of Ripon, the Marquis and Marchioness of Stafford, the Marquis of Cholmondeley, the Marquis of Northampton, Maria Marchioness of Ailesbury, the Dowager Marchioness of Conyngham and Lady Blanche Conyngham, the Marchioness of Downshire, the Countess of Rosslyn, Earl and Countess Compton, the Earl of Feversham, the Earl of Durham, the Countess of Coventry and the Ladies Coventry, Countess Howe and Ladies Curzon, the Countess of Yarborough, the Earl of Clonmell, the Earl and Countess of Loudoun, the Countess of Kilmorey, Countess de Grey, Viscount Molyneux, the Bishop of Gibraltar and Mrs Sandford, Viscountess de Labrosse and Mdles. de Labrosse, Lord Greenock, Lord and Lady Algernon Lennox, Lady Lilian Wemyss, Lady Julia Wombwell and Miss Wombwell, Hon. Lady Campbell of Blythswood, Lady Constance Harris, Lady Florence Harris, Lord and Lady Burghley, Lady Dorchester, Lord Houghton, Lady Curzon, Lady Brougham and Vaux and Miss Musgrave, Baron de Hirsch, Helen Lady Forbes and the Misses Forbes, Baroness Burdett Coutts and Mr Burdett Coutts, M.P., Lady Astley, Lord and Lady William Nevill, Lord and Lady Brooke and Hon. Leopold and Hon. Marjorie Greville, Lady Dalton Fitzgerald and Miss Payne, Lord Elcho, Lady Willoughby de Eresby and the Misses Willoughby, Lady Ewart, Lady Hindlip, Lady Cecilia Rose, Lord Rowton, Lord Capell, Lady Mabel Compton, Lord and Lady Herries, Lady Sykes, Lord and Lady Alwyne Compton, Lady Ventry and Hon. Miss de Moleyns, Hon. Lady Knollys, Lady Manners, Hon. Lady Drummond and Misses Drummond, Hon. Sydney Greville, Hon. Assheton Harbord, Hon. Derek Keppel, Hon. Captain Bourke, Hon. Mrs Percy Mitford, Hon. Philip Harbord, Hon. and Rev. L. Tyrwhitt, Hon. Coralie Glyn, Hon. M. Lambton, Hon L. Greville, Hon. Alwyn Greville, Hon[.] Eva Greville, Hon. Frances Dawnay, Hon Evelyn Paget, Hon Mrs Algernon Grosvcnor, Hon and Mrs Lambton. Hon. Mrs Evelyn Pelham, Hon. Fitzroy St Clair Erskine, [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], Hon. W. Maxwell, Hon. G. and Mrs Dawnay, Hon. Captain Lambton, General Sir Seymour Blane, Sir William Gordon Cumming, Sir Archibald and Miss Edmunstone, Sir Henry and Lady Meysey Thompson, Mrs Gladstone, Mr and Lady Margaret Cameron, Mr and Lady Florence Astley, Mr and Lady Elizabeth Melville Cartwright and Miss Melville Cartwright, General and Mrs Duff and Miss Duff, General Owen Williams, Colonel and Mrs Anstruther Thompson, Colonel Stopford Sackville, Colonel Craig, Colonel Brabazon, Colonel Forester, Colonel Stanley Clarke (in attendance on the Prince of Wales) and Miss Clarke, Major Davidson, Canon and Mrs Fleming, Captain St John Mildmay (in attendance on the Duke of Cambridge), Captain and Mrs FitzGeorge, Mrs William Portal, the Sub Dean of the Chapels Royal, Rev. W. M. and Mrs Wollaston, Rev. R. West, Mr and Miss Chamberlain. Mr and Mrs W. H. Grenfell, Mr Christopher Sykes, M.P., Mr Reuben Sassoon, Mr Arthur Sassoon, Mrs Bischoffsheim, Mr and Mrs Herbert Naylor Leyland, Dr and Mrs Anderson, Mrs Oswald, Mr and Mrs Surtees, Mr Fitzgerald, Mr and Mrs Menzies, Miss Willoughby and Miss Gertrude Willoughby, Mr Arthur Paget, Miss Christie, Mr Dowdall, Mrs Eastwood and the Misses Eastwood, Mr Frederick Warre, Mr William Williamson, Mrs Burton, Mrs Chaine, Captain and Mrs Livesey Wardle, Captain A. R. Hill, Miss Helby, Mrs and Miss Winslow, Mr and Miss Oswald, Mr Willoughby, Mr Quintin Hume, Mr F and Miss Fitzherbert, Mrs and Miss F. Hotham, Mr and Mrs Moreton Frewin, Mrs Charles Otway, Mr Calcraft, Mr Watson, Mr Fennick, Mr Montgomery, Mr Cecil Murray, Mr and Mrs Sneyd, Mr Leslie Stephen, Mr Edgar Sebright, Miss Fleetwood Wilson, Mrs and Miss Farmer of Nonsuch, Mr and Mrs Dennistoun and Miss Dennistoun, Mr and Mrs J. R. Lane Fox and Miss Lane Fox, &c. The bride and bridegroom left early in the afternoon for Easton Lodge, Dunmow, the seat of Lord and Lady Brooke, in Essex, where they will spend the honeymoon. The bride travelled in a costume of biscuit coloured armure silk, the top of the bodice and sleeves being of pale pink mousseline de soie, and the lower part of the former inserted with pink velvet ribbons to form a high band, Tuscan straw-hat trimmed with pink crepe de chine, and biscuit feathers. The presents exceeded four hundred in number. The Prince of Wales presented the bride with a trefoil moonstone and diamond brooch and the bridegroom with a sapphire and diamond pin. The Princess of Wales's gift the bridegroom was a silver-necked claret jug. The Duke of Cambridge gave an amethyst and diamond pendant, and the bridegroom a pair of silver gilt candlesticks; and Princess Mary and the Duke of Teck presented the bride with a silver sugar basin. The bridegroom’s gifts to the bride were a diamond and sapphire watch bracelet, a diamond half-hoop ring, diamond brooch with pearl centre, diamond and turquoise brooch, formed of the St Clair badge in rubies and diamonds set in a circle of diamonds, surmounted by a baron’s coronet in the same precious stones, a large salts bottle with two heart-shaped moonstones set with rubies and diamonds and a silver sugar basin. The bride presented Lord Loughborough with a turquoise and diamond stud. The Earl of Rosslyn presented the bride with diamond necklet. The Countess of Rosslyn’s gifts were diamond orchid brooch and a pink feather fan. Mr R. C. Vyner’s gifts to his daughter were a diamond and pearl tiara and gold mounted dressing case, and Mrs Vyner gave her a sapphire and diamond ring. The Marchioness of Stafford gave a ruby and diamond pin; Lady Mary Vyner antique diamond tiara; the Ladies Angela and Sybil Erskine sapphire and diamond bracelet; Hon. Fitzroy St Clair Erskine with "1890" in rubies and diamonds; Lord and Lady A. Compton large diamond star; the Earl of Rosslyn presented his son with a large silver bowl and two silver dishes; the Countess of Rosslyn's present was a travelling bag; Mr Vyner gave him a luncheon basket; Mrs Vyner a silver ''entree'' dish; the Marchioness of Stafford case containing silver coffee and tea pots, urn and stand, sugar basin, and cream jug; Lord and Lady Brooke four antique silver sauce boats; Lord and Lady Algernon Gordon ['''Col. 1c–2a'''] Lennox three antique silver ''entree'' dishes. The Earl of Rosslyn's Fifeshire tenants presented Lord Loughborough with a beautiful antique silver bowl, two antique silver dessert dishes, and two antique sugar castors; and to the bride they gave an antique silver tea service. The congregation of Rosslyn Chapel presented the bridegroom with a complete and elegant writing set, whilst Lord Rosslyn's servants gave him a silver inkstand and pair of silver candlesticks. Mr Vyner's servants presented the bride with a silver salver and six ''liqueur'' cups. The servants of Newby Hall gave a silver coffee pot, and Mr Vyner’s Cheshire tenants presented a case containing silver tea pot, sugar basin, and cream jug. The bride also received from the Duchess of Portland a travelling bag; the Duchess de Luynes, sapphire and diamond ring; the Duke of Westminister, Indian necklace; the Marquis of Northampton, diamond bracelet; the Marquis of Ripon, sapphire and diamond bracelet; the Marchioness of Ripon, turquoise and diamond bangle, the Earl of Durham, a diamond half hoop bracelet; Earl and Countess Cowper, a diamond "V V” brooch; Dowager Viscountess Downe, moonstone and diamond crescent; Lady Skyes [Sykes] ruby and diamond fish brooch; Lady Millbank, sapphire and diamond brooch Lord Douglas Compton, double heart brooch; Lord Herbert Vane Tempest, gold curb chain bracelet set with pearls; Earl and Countess Compton, bangle with “Vi” and “good luck” on it; the Duke and Duchess of St Albans, tea service with table; the Marchioness Conyngham, silver cream jug; the Marchioness of Tavistock, silver vase; the Countess of Rodin, a silver I bowl; the Countess of Yarborough, silver candlesticks; Viscountess Downe, silver tea caddie; Lady de Trafford, gold mounted salts bottle; the Hon. F. Dawnay, large silver candlesticks[;] Hon. F and Lady Lampton, travelling clock; Hon. W. Lambton, silver candlesticks; Sir R. and Lady Graham, silver salver; Sir Roderick and Miss Cameron, two silver dishes; General Williams, diamond bangle; Colonel and Mrs G. FitzGeorge, gold mounted bottle; Major and Mrs Egerton, pair of silver candlesticks; Major Stuart Wortley, diamond Swallow; Major Davidson, heart-shaped brooch in coral and diamonds; Captain and Lady Cecilia Rose, a brougham; Mr Beaumont, silver bowl; Miss Edmonstone, moonstone and diamond hat pins; Mrs Harris, sapphire and diamond bracelet; Mr Egerton, turquoise and diamond brooch; Sir Cecil Murray, sapphire and diamond pin; Mr A. Farquharson, diamond swallow brooch; Mr and Mrs Hulse, two small silver vases; Miss Fleetwood Wilson, diamond bangle; Mrs A. Wilson, tortoiseshell candlestick; Mr Quintin Hume, salts bottle mounted with diamonds and rubies; Miss Helbert, gold violet pin; Mr Chamberlain, pair of silver candlesticks; Hon. Arthur Coventry, gold mounted bottle with "Vi" in diamonds; Mrs Leopold de Rothschild, diamond hand brooch, together with a number of other gifts. Among the presents to the bridegroom were — from the Marchioness of Downshire, antique silver sugar sifter; the Marquis of Exeter, silver sandwich case; the Marchioness of Bath, antique Dutch silver box; the Marquis of Downshire, a gold pencil; Count Munster, cigarette case and match box; Count Alexander Munster, a hunting flask; the Countess of Lindsay, antique silver pen-tray; the Earl of Mar and Kellie, silver matchbox; the Earl and Countess of Loudoun, silver cigarette box; the Earl of Chesterfield, antique silver gilt inkstand; the Countess of Bradford, gold seal with agate; the Countess of Warwick, antique silver bowl; the Earl of Coventry, silver pen-tray; the Countess of Wharncliffe, two silver candlesticks; the Earl and Countess of Cromartie, two antique silver cups; Viscountess of Newport, silver paper knife; Mr and Mrs A. Sassoon, ruby and diamond pin; Lord John Cecil, silver salver; Lord and Lady Willoughby de Eresby, antique silver-mounted cruet stand; Lord William Nevtll, antique brandy carrier; Lady Margaret Spicer, silver match box; Lord Burghley, four silver salt cellars and spoons; Baron de Hirch [Hirsch?], silver inkstand; Lady W. E. Osborne Elphinstone, silver inkstand; Lord Kesteven, silver cream jug; Lord Capell, silver cigarette case; Lord and Lady Gerard, tortoiseshell and silver travelling clock; Lady Lilian Wemyss, silver cigarette case; Dad Alwyne Compton, umbrella containing matchbox and whistle; Lord and Lady Lennox, two antique silver ''entree'' dishes and bacon dish; Lord Burghersh, silver cigarette lighter; Lady Angela 8t Clair Erskine, four silver salt cellars; Lady Randolph Churchill, gold pencil case, with two sapphires; Lord Houghton, antique gold basket; Lady Sybil St Clair Erskine, umbrella; Lord and Lady Dorchester, large crystal seal; Hon. Derek P. Keppel, pair of silver candlesticks; Hon. E. and Mrs Pelham, silver inkstand; Hon. G. N. Curzon, silver gilt bowl; Hon. F. St Clair Erskine, four silver candle-sticks; [[Social Victorians/People/Bourke|Hon. A. and Mrs Bourke]], four small silver dessert dishes; Hon. Mrs Elliot Yorke, pair of silver candlesticks; Hon. Mr and Mrs L. Greville, tortoise-shell and silver box; Hon. R Somerset, diamond and pearl pin; Hon. G. H. Willoughby, silver ink-stand; Hon. H. and Mrs Grosvenor, umbrella; [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], enamel and diamond pin; Captain Hon. Maurice Bourke, cigarette case and lighter; Hon. W. Maxwell, silver-backed clothes and hat brushes; Hon. Sidney Greville, saltspoons; Hon. M. C. Maxwell Scott, two silver ''menu'' plates; Hon. W. Edwardes, pin; [[Social Victorians/People/Greville|Captain Hon. A.]] and [[Social Victorians/People/Lady Violet Greville|Mrs Greville]], silver gilt bowl, fork, spoon, and ladle; Sir Frederick Milner, diamond and moonstone pin; Sir George Chetwynd, two long silver flagons; Sir M. and Lady Fitzgerald, two silver saltspoons; Sir N. Gordon Gumming, silver-handled paper knife, &c.<ref>"Marriage of Lord Loughborough with Miss Vyner." ''Fife Free Press'' 26 July 1890, Saturday: 2 [of 8], Col. 1a–2b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001110/18900726/015/0002. Same print title and p.</ref> </blockquote> === 24 July 1890, Thursday === ==== Dinner and Dance Hosted by Lord Alington ==== Perhaps the date on this is uncertain, since the article in ''Truth'' would suggest it was held in late May or early June. The newspaper story in the ''Queen'' mentions another occasion in which the flowers were notable as well. From ''Truth'':<blockquote> But not a quarter of a mile distant [from a dinner and dance hosted by Lord Rothschild featuring Princess Louise and the Marquis of Lorne] a more important event was proceeding, for Lord Alington, having previously walked down the Row and invited his friends right and left to Alington House, South Audley-street, proceeded to give a dance which was honoured by the presence of the Prince and Princess of Wales, the Princesses Victoria and Maud, the Duke of Clarence, Princess Mary Adelaide, and Princess "May" and Prince and Princess Christian. This without exaggeration, with the exception of Lady Leconfield's, Lady Willoughby de Eresby's, and Lady Cadogan's, was the best ball hitherto given this season, and the prolific way ortolans [small songbirds regarded as a delicacy] were distributed at supper will long linger in the memories of those by whom these delicacies are cherished. One lady displayed her enthusiastic appreciation by devouring five of these, and it is to be hoped her devotion did not endanger her life. As for beauty, seldom before have so many pre-eminent for good looks been present at one gathering, as a mere enumeration of a few of these will prove. Of course the Duchess of Leinster heads the list, followed in quick succession by the Duchess of Portland, Lady de Grey, Lady Randolph Churchill, Lady Dudley, Lady Kilmorey, Lady Brooke, Lady de Trafford, Lady Helen Duncombe, Lady Hilda Dundas, Lady Sybil St. Clair Erskine, Lady Olivia Taylour, Miss Rachael Gurney, Lady Cardross, Miss Bourke, Lady Rose Leigh and her sister-in-law, Miss Leigh, Miss Berens (sister to Lady Cairns), Mrs. Ralph Sneyd, Lady Gerard, and a host of others far too numerous to mention. I often wonder that no painter has yet attempted to give us in one picture a permanent record of the chief beauties of our day. It would make a most interesting and exquisite group of lovely women, such as could not be found probably in any other country than this.<ref>Marmaduke, pseud. "Letters from the Linkman." ''Truth'' 12 June 1890, Thursday: 26 [of 56], Col. 1a–b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18900612/064/0026. Same print title, p. 1212.</ref></blockquote> From the ''Queen'':<blockquote>The P<small>RINCE AND</small> P<small>RINCESS OF</small> W<small>ALES</small> and the Princesses Victoria and Maud honoured L<small>ORD</small> A<small>LINGTON</small> by their presence at D<small>INNER</small>, at A<small>LINGTON</small> H<small>OUSE</small>, South Audley-street, on the 24th ult., when a distinguished party was present to meet them, including, among others, the Duke and Duchess of Leinster, the Marquis of Hartington, the Marquis and Marchioness of Ormonde, Maria Marchioness of Ailesbury, Earl and Countess Cadogan and Lady Emily Cadogan, the Earl Countess of Lucan and Lady Rosaline Bingham, the Earl of Hardwicke, the Countess of Dudley and Lady Edith Ward, the EarI of Chesterfield, the Earl of Stradbroke and Lady Hilda Rous, and Earl Annesley. Covers were laid for sixty. Lord Alington afterwards gave a dance to meet the Prince and Princess of Wales and their daughters. As on the previous occasion, the floral decorations were exquisite. The entrance hall and the approaches to the reception rooms were decorated with tall palms and a profusion of lilies, pelargoniums, ferns in every variety, and roses in every imaginable shade. The conservatories at the back were carpeted and furnished luxuriously, and blocks of ice, lighted by fairy lamps, were placed about with artistic effect, serving to keep the atmosphere cool and fresh. Feathery palms, hydrangeos [hydrangeas] and arums were here used freely, and a particularly pretty result was obtained by the windows being framed, one in a fringe of ferns and red pelargoniums, and the other in ferns and fuschias. The ballroom fireplaces were adorned on each side with palms reaching to the ceiling, the mantelpieces being hidden by a delicate arrangement of roses in various shades springing from banks of hydrangeas, calceolarias, pelargoniums, and marguerites, and the walls of the staircase leading from the conservatory to the ball-room were hung with half-balls of floral roses. The supper was served in the dining-room, the tables being embellished with the loveliest orchids and roses of the most delicate tints, intermixed with pancretia; the arrangement was so light and elegant that it was difficult to believe that hands had ever touched the flowers. On those tables with silver candelabra in the centre, the flowers were all pure white, and those having gold candelabra were treated with mauve orchids and the palest pink and yellow roses. A tent, arranged near the conservatory for the Prince of Wales, was draped with Oriental hangings, and outside were groups of flowers and myriads of fairy lamps. The company began to arrive at eleven o'clock, when dancing commenced to the strains of the Blue Hungarian Band. The Princess of Wales was dressed in white striped satin and silk, the front being embroidered in pale shades, and wore handsome diamond ornaments. The Princesses Victoria and Maud wore white tulle over satin, the front being sprinkled all over with tiny sprays of lilies of the valley, each laid on a single leaf. The Duchess of Wellington was in black satin, with a pink sash arranged very prettily across the back of the bodice, and falling to the feet; a cluster of pink flowers and a beautiful diamond spray were fastened in the front of the bodicea and a diamond rivière was arranged in the hair, a diamond necklace being also worn. The Duchess of Westminster was in white satin trimmed with gold embroidery, and veiled in white chiffon. The Marchioness of Ormonde wore white striped brocade and satin trimmed with white lace, and a splendid diamond tiara and other ornaments. Countess de Grey, palest blue satin, with front beautifully embroidered in beads, and a diamond and pearl coronet. Maria, Marchioness of Ailesbury, white brocade and lace, with splendid diamond and emerald ornaments. The Marchioness of Bath, cream an violet brocaded velvet, the front being covered with mauve [38, Col. 3c – 39, Col. 1a] violets. Lady Mary Mills, rose-pink satin, with front richly embroidered; Lady Cardross, white satin, trimmed round the bodice and train with silver embroidery; Lady William Nevill, pale yellow brocade, the front veiled with black net, trimmed up the side with jet leaves, and all round the skirt was a bordering of Maréchal Niel roses, over a flounce of black net; Countess Granville wore rich white brocade and beautiful jewels; the Countess of Sefton, black satin and diamonds; Viscountess Dungan, white mousseline de soie, the skirt edged with a deep flounce; Lady Brooke, pale blue satin, trimmed with silver; Lady Emma Crichton, white satin train, with front of white lisse, prettily embroidered in red; Lady Idina Brassey, pale mauve bengaline, with front of chiffon and a handsome diamon coronet; Lady Rose Molyneux, white cross crêpe de Chine, with a pale green sash, and prettily trimmed with pink roses; Lady Wimborne, pale grey satin; the Hon. Miss Guest, pink satin, veiled with white net, sparkling with silver; Lady Sarah Spencer Churchill, white satin, with a chiffon frill; Lady Chesham, white silk, with pink roses; the Countess of Normanton, pale grey satin, with front of fine old lace; Mrs Hulse, handsome white brocade, diamond tiara, &c.; Mrs Bischoffsheim, white silk, embroidered with silver, and trimmed round the bottom of the skirt and train with a ruche of pale yellow crèpe; the Hon. Muriel Brassey, white chiffon, over satin, slightly trimmed with pink; and the [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]] was in pink, with magnificent diamonds.<ref>"Entertainments, Balls, &c." ''The Queen'' 2 August 1890, Saturday: 38, Col. 3c – 39 [of 82], Col. 1a–1c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18900802/237/0038 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18900802/237/0039. Print: ''The Queen, The Lady's Newspaper'', pp. 170–171.</ref></blockquote>This is the story from the ''Morning Post'':<blockquote>The Prince and Princess of Wales and the Princesses Victoria and Maud honoured Lord Alington by their company at dinner last evening at Alington House, South Audley-street. The guests present to meet their Royal Highnesses comprised the Duke and Duchess of Leinster, the Marquis of Hartington, the Marquis and Marchioness of Ormonde, Maria Marchioness of Ailesbury, Earl and Countess Cadogan and Lady Emily Cadogan, the Earl and Countess of Lucan and Lady Rosaline Bingham, the Earl of Hardwicke, the Countess of Dudley and Lady Edith Ward, the Earl of Chesterfield, Earl Annesley, the Earl of Stradbroke and Lady Hilda Rous, Lord and Lady Gerard, Lady Norreys, Lord Rowton, Lord Calthorpe, Lord Frederick Hamilton, Lord Charles Montagu, Lord Herbert Vane Tempest, Lord Arthur Grosvenor, Lady Lilian Wemyss, Colonel the Hon. Oliver Montagu, Sir John and Lady Lister Kaye, the Hon. Humphrey and Lady Feodore Sturt, the Hon. John Baring, the Hon. Henry and the Hon. Julia Stonor, the Hon. Gilbert Willoughby, Lady Suffield and Major-General Ellis, in waiting on the Prince and Princess of Wales; Sir Henry Calcraft, Mr. and Lady Rose Leigh, General and Mrs. Owen Williams, Mr. Mildmay, M.P., Mr. and Mrs. Arthur Sassoon, Mr. Montague Guest, Captain Orr-Ewing, Mrs. Bulkeley, Mr. Ricardo, and Mr. Caryl Craven. Lord Alington afterwards gave a dance expressly to meet the Prince and Princess of Wales. Among the company were— The Russian Ambassador, the Danish Minister, Prince Louis Esterhazy, Comte Costa de Beaunyardt, the Duchess of Buccleuch and Lady Evelyn Fitzmaurice, the Duchess of Westminster, the Dowager Duchess of Marlborough and Lady Sarah Spencer Churchill, the Duchess of Wellington and Lady Cardross, the Marquis and Marchioness of Granby and Miss Bourke, the Marchioness of Stafford and Lady Sybil St. Clair Erskine, the Marchioness of Bath and Lady Beatrice Thynne, the Marquis of Cholmondeley, the Marquese and Marquesa de Santurce and Lady Wilham Nevill, Countess Spencer, Countess Granville and Lady Victoria Leveson-Gower, Countess de Grey, the Earl of Dunraven, the Countess of Lathom and Lady Edith Wilbraham, the Countess of Enniskillen and Lady Christian Cole, the Earl of Ilchester, the Earl of Durham and Lady Anne Lambton, the Earl of Dalkeith and Lord Ernest Scott, the Countess of Sefton and Lady Rose Molyneux, Countess Howe and the Ladies Curzon, the Countess of Normanton and Lady Mary Agar and Miss Mary Liddell, Earl Craven, Viscount Clifden, Viscount Ennismore and Lady Beatrice Hare, Viscount Coke and the Hon. Violet White, Viscount and Viscountess Castlerosse and the Hon. Susan Baring, Viscountess Newport and the Hon. Miss Bruce, Viscount and Viscountess Melgund, Viscount and Viscountess Curzon, Viscount and Lady Violet Dangan and Lady Eva Wellesley, Viscount Dungarvan, Viscount Chelsea, Viscount Royston, Lady Hastings, Lord Capell, Lord and Lady Willoughby de Eresby and Hon. Margaret Willoughby, Lady Cloncurry and Hon. Miss Winn, Lord and Lady Brooke, Lady Wimborne and Hon. Miss Guest, Lord Sandhurst, Lord and Lady Edmund Talbot, Lord and Lady Alwyne Compton and Miss Gurney, Lord and Lady Chesham, Lady Rossmore and Miss Naylor, Lord Kenyon, Lord Garioch, Lady Alice Portal, Lady Margaret Levett and the Ladies Ashley, Lady W. Nevill, Lord Paulet, Lady Charles Beresford, Lady Forbes and Misses Forbes, Lady Claud Hamilton, Lord Carlow, Lady Mary Mills, Lord Lurgan, Lord Henry Vane Tempest, Lord and Lady Algernon Gordon Lennox, Lord Richard Nevill, the Hon. W. Coventry, the Hon. G. Joliffe, the Hon. Schomberg M'Donnell, the Hon. H. Coventry, Captain the Hon. E. Dawson, the Hon. Alistair and Mrs. Hay, the Hon. Mrs. Grosvenor, [[Social Victorians/People/Bourke|the Hon. Algernon and Mrs. Bourke]], the Hon. F. Dawnay, the Hon. E. J. Mills, the Hon. A. Johnstone, the Hon. Claude Hay, the Hon. Sidney Greville, the Hon. Mr. Dundas and Lady Hilda Dundas, the Hon. Thomas and Lady Idina Brassey and the Hon. Muriel Brassey, the Hon. Mr. Mansfield, Colonel the Hon. Henry Byng and Miss Byng, the Hon. Mrs. Oliphant, the Hon. Mr. Harbord, the Right Hon. Henry Chaplin, M.P., the Hon. G. Somerset, Sir William Gordon Cumming, Sir George Arthur, Sir Hubert Miller, Sir Charles Forbes, Sir William and Lady Eden, Sir George Chetwynd, Sir Frederick Johnstone, Sir John D. Poynder, Colonel and Lady Emma Crichton, the Hon. Miss Harbord, Captain and Lady Jane Combe, Colonel Paget, Captain Williams, Captain Ford, Captain Doyle, Captain Hanbury, Captain Campbell, Captain Constable, Captain Kinloch, Captain Longfield, M. Gutmansthal, Mrs. Bingham, Mr. Guthrie, Mrs. Arthur Paget, Mrs. Charles Wilson and Miss Wilson, Mr. Hulse, M.P., and Mrs. Hulse, Mr. de Murrieta, Mrs. Bischoffsheim, Mr. Spencer Portal, Mr. and Mrs. Gerald Paget, Mr. Leigh and Miss Leigh, Mr. Percy Daniell, Mr. Lister, Mr. Frederick Vincent, Mr. Cazalet, Mr. Alfred de Rothschild, Mr. John Savile Lumley, Mr. and Mrs Leopold de Rothschild, Mr. and Mrs. Skeffington Smith and Miss Taylor, Mrs. George Cavendish Bentinck, Mr. Alfred Farquhar, [[Social Victorians/People/Fanny Ronalds|Mr. Reginald Ronalds]], Miss Alice Ellice, Mr. George Marjoribanks, Mr. Wyndham, Mr. Dugdale, Mr. Montagu, Mrs. Sneyd, Mr. Wynn, Mr. Edward Hamilton, Mr. Murray Guthrie, Mr. George Lambert, Mrs. Sanford, Miss Mabel Forbes. Mr. Herbert Magniac, Mr. Bailey, Mr. Milner, Mr. Charles Kindersley, Mr. Beaumont, Mr. Walter Hervey, Mr. H. Hoare, Mr. Foley, Mr. Scott Murray, Mr. Coventry, Mr. Bertie, Mr. Jarvis, Mr. Guthrie, Mr. George Cavendish Bentinck, Mr. E. Charteris, Mr. Cholmondeley, Mr. Oswald Magniac, Mr. Cecil Campbell, Mr. and Mrs. Henry Oppenheim, Mr. Peploe, Mr. A. Charteris, Mr. Seymour Gore, Mr. Robert Antrobus, Mr. Elliot, M. de Champs, Mr. Wombwell, Mr. Charles Hoare, M.P., and many others.<ref>"Lord Alington's Ball." ''Morning Post'' 25 July 1890, Friday: 3 [of 8], Col. 5 b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18900725/014/0003. Same print title and p.</ref></blockquote> ==August 1890== ===6 August 1890, Friday=== ==== Beautiful Women ==== The ''Lancaster Observer and Morecambe Chronicle'' reprinted a story from ''Vanity Fair'' about "Beautiful Women":<blockquote>Beautiful Women. — ''Vanity Fair'' asks what country in the world can show a more beautiful bevy of women than England? When Lady Helen Vincent, with her sister, Lady Cynthia Graham; when Lady Warwick with her sisters, the Duchess of Sutherland and Lady Westmorland; and when Lady Mar and Kellie, Lady de Trafford, Princess Pless, Lady de Gray[,] Lady Naylor-Leyland, the Duchess of Portland, Lady Alington, Lady Sophie Scott, Lady Chelsea, [[Social Victorians/People/Bourke|Lady Algernon Bourke]], Lady Henry Bentinck, Mrs. Willie Grenfell, Mrs. Rupert Beckett, Mrs. Arthur Paget, and Countess Clary are gathered together it is a sight to make an old man young; while Mrs. Gerard Leigh and her sister, Lady Yarborough, and Lady Powis, Lady Norreys, Mrs Farquharson, Mrs. Willie James, Lady Dalkeith, Mrs. Pretyman, Mrs. Gervase Beckett, Lady Rossmore, Lady Sarah Wilson, Lady Colebrook, Lady Hartopp, Mrs. Willie Walker, and Lady Milton are almost equally admired. The many pretty girls include Miss Enid and Miss Muriel Wilson, who still head the list of the unmarried beauties, Lady Beatrice Butler, Lady Alexandra Hamilton, Miss Kennard, Miss Madeline Bourke, and Miss Leila Crichton; while among the debutantes Lady Evelyn Crichton and Lady Vivian's daughters have had a great success, and Lady Alice Montagu, petite and charmingly pretty, has been so much admired. The daughters of Lady Gosford, Lady Verulam, Lady Leitrim, Lady Inchiquin, and Lady Grey have also been admired, as well as Lady Helen Stewart, Lady Katherine Scott, and Miss Chaplin.<ref>"Beautiful Women." ''Lancaster Observer and Morecambe Chronicle'' 6 August 1897, Friday: 2 [of 8], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001804/18970806/023/0002. Same print title and p.</ref></blockquote> ===25 August 1890, Monday=== Summer Bank Holiday ==September 1890== ==October 1890== ===22 October 1890, Wednesday=== Arnold Dolmetsch went to the Reading Room (Campbell 22). ===25 October 1890, Saturday=== [[Social Victorians/Loder De Vere Beauclerk Wedding|Loder—De Vere Beauclerk Wedding]] ===31 October 1890, Friday=== Halloween. ==November 1890== ===5 November 1890, Wednesday=== Guy Fawkes Day ===19 November 1890, Wednesday=== Arnold Dolmetsch, with some of his students and his daughter Helene, provided musical illustrations for a lecture given by Dr. Frederick Bridge on "Mozart as Teacher," probably at Gresham College, where Bridges was professor (Campbell 34). ===21 November 1890, Friday=== Dr. Frederick Bridge's last fall 1890 lecture at Gresham College, for which Dolmetsch provided illustration. Campbell says, "it is, in fact, the first record in modern times of music by Byrd, Dr. Bull, Jenkins, Simpson, Lawes, Locke and Purcell being played 'on the instruments for which it was written'" (Campbell 34). ===25 November 1890, Tuesday=== Arnold Dolmetsch got a three-month pass to the Reading Room and "thereafter held yearly tickets until 1916" (Campbell 22). ==== 1890 November 29, Saturday ==== ===== [[Social Victorians/Dudley-Beckwith Wedding 1890-11-29|Wedding of Francis Dudley, Lord Leigh and Héléne Forbes Beckwith]] ===== ==December 1890== Sometimes in December 1890, [[Social Victorians/People/William Robert Woodman|William Woodman]], one of the founding members of the [[Social Victorians/Golden Dawn|Golden Dawn]], died (Gilbert 86 12). === 4 December 1890, Thursday === ==== [[Mure-Portal Wedding 1890-12-04|Wedding of Miss Mure and Mr. S. J. Portal]] ==== ===19 December 1890, Friday=== "The eleventh annual ''Truth'' Toy Exhibition for the children in London hospitals, workhouses, and infirmaries was held at the Grosvenor Gallery, Bond-street, on Dec. 19 and 20. There are over 22,000 children in these institutions, and each of them has a separate toy, besides the large and more expensive toys which are presented for the general use of the inmates. A special feature of the show was the dolls dressed by lady readers of ''Truth''. Several special donations in connection with the fund have been made, the principal of which is that of 10,200 new sixpences sent for distribution by the same liberal donor who has given a similar present on former occasions; while Mr. Tom Smith has had specially made, and has forwarded for distribution, over 22,000 crackers; and Mr. Horniman (as stated in our last issue) has given to the adults in hospitals and workhouses 1200 packets of tea, to be given to them as Christmas boxes." (Illustrated London News (London, England), Saturday, December 27, 1890; pg. 806; Issue 2697, Col. B) ===20 December 1890, Saturday=== "In order that the adults connected with the hospitals and workhouses may participate in the good things given to the children in the ''Truth'' toy distribution, Mr. F. J. Horniman, of Wormwood-street, City, has sent Mr. Labouchere 1200 packets of tea to be given to them as a Christmas box." (Illustrated London News (London, England), Saturday, December 20, 1890; pg. 774; Issue 2696, Col. B) From the ''Times'', 22 December 1890, "The Late Sir Edgar Boehm": <blockquote>On Saturday, in accordance with the special request of the Queen, the late Sir Edgar Boehm, R.A., was buried in St. Paul’s Cathedral, in the south aisle of the crypt, the spot being known as “The Painters Corner.” [new paragraph] The funeral cortége left the residence of the deceased at about 11 o’clock, there being five mourning and several private carriages. In the first carriage were the chief mourners, Sir Edgar Collins Boehm, the only son of the late baronet, and Mr. Conrad Herapath. The second and third carriages were occupied by Sir Frederic Leighton, P.R.A., Sir Nigel Kingscote, Mr. Richard Mills, Colonel Francis Baring, Mr. Alma-Tadema, R.A., Sir R. H. Collins, Mr. Edward J. Poynter, R.A., and Mr. Duncan MacGregor, all of whom subsequently, at the cathedral, acted as pall bearers. The fourth carriage contained representatives of the Council of the Royal Academy – Mr. Edwin Long, R.A., Mr. J. B. Burgess, R.A., Mr. F. Goodall, R.A., and Mr. Beavis. In the fifth carriage were the following members of the professional staff of the deceased – Mr. R. Glassby, Mr. E. Lantéri, Signor Finili, and Herr Gross. The cortége stopped for a few moments at the Royal Academy, in order that it might be joined by two carriages containing other representatives of the institution. Leaving Piccadilly, the route taken was Waterloo-place, Pall-mall, Cockspur-street, the Strand, Fleet-street, and Ludgate-hill, the cathedral being reached about noon. The procession was met at the great west door by Canon Gregory, the Archdeacon of London, the Rev. Dr. Baker, Minor Canons Milman, Russell, and Kelly, and the choir, who had a few minutes previously left the vestry; by Major Bigge, Equerry to the Queen, representing her Majesty; Colonel Stanley Clarke, who attended on behalf of the Prince and Princess of Wales; and Colonel W. J. Colville, representing the Duke of Edinburgh. The [[Social Victorians/People/Princess Louise|Princess Louise]] (Marchioness of Lorne) attired in deep mourning, and accompanied by [[Social Victorians/People/Arthur Collins|Lieut.-Col. Arthur Collins]] and Lady Sophia Macnamara, arrived at the cathedral a few minutes before 12 o’clock, and was conducted to a seat in the south side of the nave, close to the late Sir Edgar Boehm’s two daughters, upon whom her Royal Highness had called earlier in the morning. As the procession came slowly up the nave towards the chancel, “I am the resurrection of life” and other sentences of the burial service were chanted to music by Dr. Croft. The coffin, which was completely hidden by splendid wreaths, was preceded by the choristers and the clergy. Then followed Sir E. C. Boehm, with Mr. Conrad Herapath and Mr. Alfred Gilbert, A.R.A., the officers and council of the Royal Academy, and members of the deceased’s studio. The coffin was deposited upon a bier, which had been placed under the dome in front of the opening into the crypt, through which the body was subsequently lowered. The mourners having been conducted to the seats which had been reserved for them, and the clergy and choir having proceeded to their places in the chancel, Psalms xxxix. and xc. Were sung to music by Purcell and Felton, followed by Spohr’s anthem, “Blest are the departed,” and the lesson 1 Cor. Xv. 20, which was read by Canon Gregory. Whilst the hymn “Days and moments quickly flying” was being sung the Archdeacon of London and Minor Canon Milman proceeded from the chancel and took up positions by the side of the coffin. The remainder of the service preceding and following the lowering of the body into the crypt was read by Minor Canon Milman, and the collect was impressively recited by Archdeacon Sinclair. The late Dr. Newman’s beautiful hymn, “Lead kindly light,” was afterwards sung by the choir, and the service was brought to a close by Beethoven’s Funeral March, which was played in compliance with the special request of her Majesty, at whose desire also Chopin’s Funeral March had preceded the service. Dr. Martin was the organist. While Beethoven’s march was being played the son and daughters of the late “Sculptor in ordinary to her Majesty,” together with other mourners, descended the crypt by the door near Howard’s monument, and saw the coffin for the last time. It bore the following inscription:- “Joseph Edgar Boehm. Born July 6, 1834; died at his work December 12, 1890. Thy will be done.” The coffin had already been deposited in the brick grave prepared for it, which is next to that of Sir Edwin Landseer. Immediately around it lie the remains of painters and sculptors of bygone days – Sir Joshua Reynolds, James Barry, Benjamin West, Henry Fuseli, Sir Thomas Lawrence, J. M. W. Turner, and John Henry Foley, while within half-a-dozen yards is the tomb of the great architect of the cathedral, Sir Christopher Wren. The coffin as last seen was covered by the floral tributes of loving friends, the symbolic laurel wreath forwarded by her Majesty surmounting them. This bore the inscription, “A tribute of gratitude for many beautiful memorial works executed for her. From Victoria, R. I.” Attached to the beautiful wreath sent by the Prince and Princess of Wales were the words, “A token of sincere regard and friendship.” Wreaths were forwarded by other members of the Royal Family, by Sir John and Lady Millais, Lord and Lady Reay, the Countess Sydney, Lord and Lady de Vesci, Mr. Henry Irving, Mr. and Mrs. Bancroft, by members of the council, students and officers of the Royal Academy, the Society of Artistic Sculptors (Vienna), and by the London Hungarian Association." (http://glassby.com/Chapter1.htm) </blockquote> ===24 December 1890, Wednesday=== "The people of South London and adjacent suburbs have to be grateful to Mr. Frederick Horniman, a well-known City merchant residing at Dulwich, for a munificent and beneficial Christmas gift. This gentleman has, during thirty years, formed a collection, worth above £100,000, of specimens of art, European, Indian, Chinese, and Japanese, also of decorative manufactures, and of ethnology and natural history, and rare books and manuscripts, at Surrey House, Forest Hill, which he intends to present to the public. The museum was opened by Sir Morell Mackenzie on Wednesday afternoon, Dec. 24. It already comprises twenty-four rooms, containing many beautiful specimens of horse and chain armour, Elizabethan bed-room furniture, an old English parlour and pantry, an Oriental armoury, a bible and manuscript room, an Egyptian gallery, and ethnographical saloon, a book and embroidery saloon, a porcelain and glass room, and a zoological saloon, with two live Russian bears and a monkey. We may describe the museum further, after another visit. Our Artist has sketched, in the ethnograhical saloon, an idol's shrine, made of cocoanut-fibre, from Fiji. Mr. Horniman, in acknowledging a vote of thanks, said that in an adjoining house he had accumulated a vast library of bibles and entomological works and specimens. It was his intention to build a hall at the back of the present building to contain a thousand persons, while the galleries would contain a library, and a stage would be provided, so that both eye and ear could be amused. He also contemplated buying sufficient land to enable a public body to erect class-rooms and gymnasia. Three cheers were given for Mr. Horniman, and the company proceeded to inspect the museum." "(Mr. F. Horniman's Museum, Forest Hill." Illustrated London News (London, England), Saturday, January 03, 1891; pg. 7; Issue 2698, Col. C) ===25 December 1890, Thursday=== Christmas Day ===26 December 1890, Friday=== Boxing Day ==Works Cited== *"Besant Chronology.pdf." From Anderson, Nancy Fix, Walter L Arnstein, Deborah Logan, and Susie L Steinbach, Volume Editors. Lives of Victorian Political Figures, Part III: Queen Victoria, Florence Nightingale, Annie Besant and Millicent Garrett Fawcett by their Contemporaries. Volume 3 of Lives of Victorian Political Figures. Series Editors, Nancy LoPatin-Lummis and Michael Partridge. London: Pickering and Chatto, 2008. *Crawford, Elizabeth. The Women’s Suffrage Movement: A Reference Guide, 1866–1928. Google Books. Accessed 14 October 2010. r8hcuxhejfdaodo77txa42kjstti6zb 0 264381 2835583 2835232 2026-10-02T22:58:50Z Scogdill 1331941 2835583 wikitext text/x-wiki [[File:Viscount Mandeville Vanity Fair 1882-04-22.jpg|alt=Old drawing for a magazine of an aristocratic man in a formal suit|thumb|"Kim," Viscount Mandeville, 1882]] == Overview == The Mandeville who was in a relationship with music-hall and vaudeville star Bessie Bellwood was Kim (George Victor Drogo) Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892). He was Viscount Mandeville from 18 August 1855 (when his grandfather died) to 22 March 1890 (when his father died).<ref name=":1" /> When he succeeded to the dukedom in 1890, he ended his relationship with Bellwood. His father was the 7th Duke and his mother was [[Social Victorians/People/Louisa Montagu Cavendish|Louise Cavendish]], "double Duchess" of [[Social Victorians/People/Manchester|Manchester]] and then [[Social Victorians/People/Devonshire|Devonshire]]. Viscount Mandeville (right) was "Man of the Day" (#255) for ''Vanity Fair'', drawn by Leslie Ward ("Spy").<ref>"Viscount Mandeville Vanity Fair 1882-04-22.jpg." Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Viscount_Mandeville_Vanity_Fair_1882-04-22.jpg (retrieved 8 September 2026).</ref> == Also Known As == *Family name: Montagu *The title Viscount Mandeville is a courtesy title for the heir apparent and eldest son of the [[Social Victorians/People/Manchester |Duke of Manchester]]. This title is one of only 2 (for the heir apparent of a duke) that is below the rank of Earl (the other is the heir apparent of the Duke of Somerset). (Is Mandeville a toponym?) *Viscount Mandeville **"Kim" (George Victor Drogo) Montagu [8th Duke], (18 August 1855 – 22 March 1890) **[[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke (22 March 1890 – 18 August 1892) **Alexander George Francis Drogo Montagu (2 October 1902 – 9 February 1947)<ref>{{Cite journal|date=2020-10-15|title=Alexander Montagu, 10th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=Alexander_Montagu,_10th_Duke_of_Manchester&oldid=983660353|journal=Wikipedia|language=en}}</ref> *Viscountess Mandeville **Consuelo Iznaga y Clement Montagu [8th Duchess] (22 May 1876 – 22 March 1890) **Helena Zimmerman Montagu (14 November 1890 – 18 August 1892) *Lord Kimbolton (Baron Kimbolton of Kimbolton is the courtesy title for the heir presumptive, the heir apparent's eldest son, so the son of Viscount Mandeville<ref>{{Cite journal|date=2026-09-21|title=Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=Duke_of_Manchester&oldid=1375987697|journal=Wikipedia|language=en}}</ref>) ** [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke (1877 – 22 March 1890) == Acquaintances, Friends and Enemies == === "Kim" Montagu, 8th Duke, Lord Mandeville until 1890 === ==== Friends ==== * Edward Russell, 24th Baron de Clifford * Derrick Westenra, 5th Baron Rossmore * Francis Needham, 3rd Earl of Kilmorey === Consuelo Iznaga y Clement Montagu === ==== Friends ==== *Consuelo Iznaga y Clement Montagu was in the intimate circle of [[Social Victorians/People/Albert Edward, Prince of Wales |Albert Edward, Prince of Wales]]. ==Organizations== === "Kim" Montagu, 8th Duke, Lord Mandeville until 1890 === * Eton<ref name=":1" /> === Consuelo Iznaga y Clement Montagu === * Consuelo Iznaga y Clement Montagu was a member of the New York City social set called the Buccaneers when she was a teenager.<ref name=":0">{{Cite journal|date=2020-10-06|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=982192002|journal=Wikipedia|language=en}}</ref> Edith Wharton is said to have take details from the 8th Duke's marriage ''The Buccaneers''.<ref name=":1">{{Cite journal|date=2020-10-06|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=982188929|journal=Wikipedia|language=en}}</ref> == Timeline == '''1855 August 18''', the 6th Duke died. The families "placed in mourning":<blockquote>He is succeeded in the Dukedom by his eldest son, Viscount Mandeville, M.P. By the melancholy event the families of the Marquess and Marchioness of Tweeddale, the Duchess of Wellington, the Duke of Montrose, Lord and Lady Olivia Ossulston [?], Lady Eliza [Elisa?] Steele, the Duke of Richmond, the Countess of St Germans, Lady Sophia Cecil, and many others of rank, are placed in mourning.<ref>"Death of the Duke of Manchester." ''Edinburgh Evening Post and Scottish Standard'' 22 August 1855, Wednesday: 2 [of 4], Col. 5c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001177%2F18550822&page=2&article=025&stringtohighlight=mandeville+viscount. Print title ''Edinburgh Evening Post and Scottish Record''; n.p.</ref></blockquote>'''1855 October 11, Wednesday''', H. I. Cridland says in a "Notice,"<blockquote>I will hold Audit Days at Lyttelton, on Wednesday, 11th October, for the receipt of rents and other monies due to the 25th of September, on account of the Soames, Jackson, Ecclesiastical, and Lyttelton Trust Estates, the lands of Lord Mandeville, Lady Emily Bathurst, Lady Olivia Sparrow, Rev. J. P. Rhoades, Rev. Charles Hodge, Dr. Merrion C. T. Maunsell, George Hart, late William Draper, and J. E. Fitz Gerald, Esqrs. ... Tenants are requested to present their last receipt.<ref>"Notice." ''Lyttelton Times'' 6 October 1855, Saturday: 1 [of 10], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003346%2F18551006&page=1&article=004&stringtohighlight=mandeville+viscount. Newspaper title ''The Lyttelton Times'', n.p. </ref></blockquote>This notice was repeated in the ''Lyttelton Times'' on 29 December 1855. '''1855 October 20, Saturday''', the [[Social Victorians/People/Wilton|Earl and Countess of Wilton]] hosted a [[Social Victorians/Timeline/1850s#1855 October 20, Saturday|house party at Heaton Hall]], with a visit to a silk mill in town. [[Social Victorians/People/Louisa Montagu Cavendish|Luise, Duchess of Manchester]] was present with the Duke and their son Lord Mandeville (Kim, later 8th Duke), who was 2 years old. '''1876 May 22''', Consuelo Iznaga y Clement and "Kim" (George Victor Drogo, 8th Duke) Montagu married in Grace Church, New York City.<ref name=":0" /> '''1890 January 28, Tuesday''', Lady Manville attended what was supposed to be [[Social Victorians/People/Lily Langtry#1890 January 28, Tuesday|Lily Langtry's opening-night performance at the St. James's]] with the Prince of Wales and some of his circle. Also on this date, the ''Dundee Evening Telegraph'' reported that Lady Mandeville [[Social Victorians/Timeline/1890#1890 January 28, Tuesday|had bought 30 "Exploration" shares]], which funded exploration expeditions with the possibility of rights to minerals or resources or the development of monopolies for goods that were "discovered."<ref>"Notes — Mainly Personal." ''Dundee Evening Telegraph'' 28 January 1890, Tuesday: 2 [of 4], Col. 2b [of my 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18900128&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''Evening Telegraph'', n.p.</ref> Her mother in law, [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess of Manchester]] also had bought 30 shares, both small amounts compared to some others. '''1890 January 29, Wednesday''', Consuelo Lady Mandeville was a guest at the house party around and the [[Social Victorians/Santurce Murrieta Wedding 1890-01-29|wedding of Miss Clarita de Murietta and the Duc de Santona]]. She had her eldest son with her, Lord Kimbolton (the 9th Duke of Manchester). Her mother in law, [[Social Victorians/People/Louisa Montagu Cavendish|Louise, Duchess of Manchester]] was also present. '''1890 February 6, Thursday''' '''(shortly before 6 February)''', Kim Lord Mandeville attended a [[Social Victorians/Timeline/1890#1890 February 6, Thursday|boxing match fought by the prize-fighter Pete Jackson]] at the New Pelican Club. Possibly related is this, from 24 January 1890:<blockquote>Further details (says the London ''Evening'' ''Post'') have transpired with regard to the fracas at the Tracadero between Charley Mitchell and Mr Abington. It seems that the cause of the row was the conspicuous and irritating production of Smith’s colours by a gentleman in the stalls in front of Mitchell. These acted upon him like the proverbial red rag, and turning to Mr Abington, he said — "You have got this up for me. I will put this cigar in your eye." Here Lord Mandeville acted the part of the peacemaker, and an adjournment was made to the bar, where the real row took place. Notwithstanding what the ''Sportsman'' says, it is a fact that "the Squire" did take off his coat to Mitchell.<ref>"The Laird of Strichen and the Prize Ring." ''Fraserburgh Advertiser'' 24 January 1890, Friday: 5 [of 8], Col. 5a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001470%2F18900124&page=5&article=035&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref></blockquote>'''1890 January 30, Thursday, through at least February 8, Saturday''', Kim Lord Mandeville was among the [[Social Victorians/Timeline/1890#1890 February 8, Saturday|recent arrivals at Brighton]] although apparently after the most notable had already left. Certainly he does not appear to have been at the party the next evening that the Prince of Wales attended. '''1890 February 9, Sunday''', the Prince of Wales came after one of Consuelo's dinner parties:<blockquote>Lady Mandeville, who has been giving a series of small dinners in her house in Cumberland Place, had a dinner on Sunday. The Prince of Wales came in after dinner to meet a small party.<ref>"Society Gossip. (From ''The World''.)." ''Leighton Buzzard Observer and Linslade Gazette'' 11 February 1890, Tuesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001091%2F18900211&page=6&article=106&stringtohighlight=mandeville+viscount. Print title: ''The Leighton Buzzard Observer'', p. 6. </ref></blockquote>'''1890 February 24, Monday''', Consuelo Lady Mandeville attended [[Social Victorians/Timeline/1890#24 February 1890, Monday|Lilly Langtry's performance as Rosalind in As You Like It at the St. James's Theatre]]. '''1890 February 25, Tuesday''', Consuelo Lady Mandeville attended a [[Social Victorians/Timeline/1890#25 February 1890, Tuesday|party hosted by Mrs. Adair]]. Viscount Mandeville is not listed among those who attended. '''1890 March 4, Wednesday''', Consuelo's brother Fernando Yznaga married New York "beauty and society leader" Miss Mabel Wright.<ref>"Personal." ''American Register'' 22 March 1890, Saturday: 5 [of 8], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003338%2F18900322&page=5&article=070&stringtohighlight=mandeville+viscount. Same print title and p.</ref> Presumably at least Consuelo was there? '''1890 March 10, Monday''', the Prince and Princess of Wales gave a [[Social Victorians/Timeline/1890#10 March 1890, Monday10 March 1890, Monday|dance at Marlborough House in honor of their wedding anniversary]]. The Viscountess Mandeville is listed among the guests invited, but the Viscount is not mentioned. '''1890 March 22, Saturday''', the Portsmouth ''Evening News'' expressed outrage about a man with a reputation like Kim's becoming 8th Duke of Manchester:<blockquote>While the House of Commons were debating the necessity for a House of Lords, the Duke of Manchester was on his death bed in Naples. There is more in the incident than appears at first sight, for Viscount Mandeville succeeds to the Dukedom, with the attendant rights of presentation to seven livings and the privilege of mutilating or rejecting Bills demanded by the people and passed by their representatives in the Commons.<p> It will hardly be urged by the moat zealous advocate of the Upper Chamber, as at present constituted, that the new Duke of Manchester, ''nee'' Viscount Mandeville, will bring much dignity, wisdom, or weight to the House of irresponsible legislators. This nobleman has had a merry time. The one sensible thing he did was to marry a charming and accomplished American lady, who, by consenting to share the name of Mandeville has kept its head above water. Viscount Mandeville himself has a weakness for bankruptcy proceedings, rows with cabmen, which music hall ladies come forward to take his part, and the class of events generally which do not approve themselves to the minds of the ordinary Briton. It is possible that Lord Mandeville may now discover that he has sown his wild oats, and may settle down with his beautiful and popular wife to the enjoyment of his  25,000 acres of land and £38,000 a-year. We have not to do with what Lord Mandeville may be. Our simple duty is to point out that it is an outrage on common decency and common sense that a man bearing the reputation of Lord Mandeville should have the right to step into the House of Lords whenever a vote is needed to swell the prejudice or sustain the selfishness that find among the Peers their natural home.<ref>"To-Day." ''Portsmouth Evening News'' 22 March 1890, Saturday: 2 [of 4], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000290%2F18900322&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''The Evening News'', n.p.</ref></blockquote>The ''Birmingham Daily Mail'' had a similar opinion, though perhaps slightly more optimistic:<blockquote>The Duke of Manchester, who expired last night at Naples, cannot be said to have been much of a public man. He led a quiet, retired life, and the announcement of his death would, if Lord  Mandeville were not heir to the title, arouse but little interest. Lord Mandeville is, unfortunately for himself, more widely known than his father was. He developed tastes for gambling, with disastrous financial results, as the details of his bankruptcy made public last year, and more recently, his name has been publicly associated with those of prizefighters and music-hall singers. Altogether, he can hardly be termed a credit to the aristocracy of England, but the higher position to which he will now attain may cause him to alter his ways and try redeem the errors of the past.<ref>"Notes and News." ''Birmingham Mail'' 22 March 1890, Saturday: 2 [of 4], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000644%2F18900322&page=2&article=027&stringtohighlight=mandeville+viscount. Print title: ''Birmingham Daily Mail'', n.p.</ref></blockquote>On the same day, Consuelo's brother and new wife were reported to apparently be immigrating to the UK:<blockquote>Among the immigrants, says a London lady correspondent, is a New York belle who has just been married to Mr. Ferdinand Yznaga, brother to Lady Mandeville and of Lady Lister-Kaye. Young Mrs. Yznaga will enter London society under the wing of her sister-in-law, Lady Mandeville, and is expected to make as great a furore as did Mrs. Langtry a few seasons ago. Mr. Yznaga married a sister-in-law of Mr. Vanderbilt, who divorced him two years ago.<ref>"An American Belle." ''Huddersfield and Holmfirth Examiner'' 22 March 1890, Saturday: 6 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003869%2F18900322&page=6&article=006&stringtohighlight=mandeville+viscount. Print title: ''The Huddersfield Examiner'', p. 6.</ref></blockquote>'''1890 November 14''', William Angus Drogo Montagu (9th Duke) and Helena Zimmerman married secretly, in London. '''1897 July 2''', the Duke and Duchess of Devonshire ( [[Social Victorians/People/Louisa Montagu Cavendish|Luise Friederike Auguste Gräfin von Alten Montagu Cavendish]] and [[Social Victorians/People/Spencer Compton Cavendish|Spencer Cavendish]]) hosted the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Lord Charles Montagu, likely Charles William Augustus Montagu, attended, as did [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]]. == Demographics == ===Nationality=== *Consuelo Iznaga y Clement was Cuban-American. *Helena Zimmerman was American. *George, William and Alexander Montagu were British. === Residences === * 1890: "The family estates — 27,312 acres, with a nominal rent-roll of £40,000 — were chiefly situated in Huntingdonshire and Armagh."<ref>"Men and Things." (London) ''Echo'' 22 March 1890, Saturday: 1 [of 4], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004596%2F18900322&page=1&article=010&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref> * In 1890 at the death of the 7th Duke, Kimbolton Castle: <blockquote>The Duke of Manchester... He was an old-fashioned Tory, and a man of simple tastes, whose chief interests were military and agricultural rather than political. Kimbolton Castle is a magnificent place, and the Duke's horses and troop of yeomanry were always the admiration of the Prince of Wales, the Marquis of Hartington, and many other visitors. Lord Hartington, it may be mentioned, was at Kimbolton when the news came of the  breaking out of the war between France and Germany [1871], he being a member of the Cabinet. The Duke had travelled much as a young man, and was familiar with the Continent, as also with South Africa, of which he was very fond. There is a private chapel at Kimbolton Castle, connected with which is a very extensive library containing some theological and historical volumes of great value. Notwithstanding his somewhat bluff manner the Duke was liked by the tenantry and poor from the kindly interest he took in their concerns. Even the poachers had a good word for his Grace. The feature, however, of the Duke's career was the marked contrast between his pursuits and those of the Duchess. The Duke loved every blade of grass, every tree and flower on his ancestral domain, and cared nothing for London life in its modern developments. Lord Mandeville [the 8th Duke] succeeds to the title, but the estates are heavily mortgaged.<ref>"Death of the Duke of Manchester." ''Pall Mall Gazette'' 22 March 1890, Saturday: 6 [of 8], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000098%2F18900322&page=6&article=014&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> == Family == *Consuelo (María Consuelo) Iznaga y Clement Montagu, Duchess of Manchester (1853 – 20 November 1909), née Doña María Consuelo Iznaga y Clement<ref name=":0" /> *"Kim" (George Victor Drogo) Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892)<ref name=":1" /> #William Angus Drogo Montagu, 9th Duke of Manchester (3 March 1877 – 9 February 1947) #Jacqueline Mary Alva Montagu (27 November 1879 – 15 March 1895). #Alice Eleanor Louise Montagu (27 November 1879 – 10 January 1900) *[[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke of Manchester (3 March 1877 – 9 February 1947)<ref>{{Cite journal|date=2020-10-08|title=William Montagu, 9th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_9th_Duke_of_Manchester&oldid=982513036|journal=Wikipedia|language=en}}</ref> *Helena Zimmerman Montagu (25 September 1878 – 15 December 1971)<ref>{{Cite journal|date=2020-10-08|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=982513630|journal=Wikipedia|language=en}}</ref> #Mary Alice Montagu Gregory (26 October 1901 – 9 October 1962) #Alexander George Francis Drogo Montagu, 10th Duke of Manchester (2 October 1902 – 23 November 1977) #Edward Eugene Fernando Montagu (26 July 1906 – 4 May 1954/1956) #Ellen Millicent Louise Montagu Hofer Shairp (5 January 1908 – 2 August 1948) === Relations === The three Iznaga y Clement sisters were prominent in London "Society": *Consuelo (María Consuelo) Iznaga Montagu, Duchess of Manchester (1853 – 20 November 1909) *Emilie Iznaga y Clement () *Natividad (Natica) Iznaga y Clement Lister-Kaye () == Questions and Notes == *In a ''British Newspaper Archive'' search for name=mandeville keyword=#viscount for 18 August 1855–22 March 1890 (September 2026), first did "Publication date oldest first," but Kim was 2 years old when his grandfather died and his father acceded to the dukedom, so on p. 57 of the search results (the page containing 1 January 1856) I switched to "Publication date newest first." Decided it would be more fruitful to work backwards. == Papers, Memoirs and Biographies == == Footnotes == {{reflist}} 1dhh5a75r1mb2miaizul6xc7r5n9vnq 2835588 2835583 2026-10-02T23:08:21Z Scogdill 1331941 2835588 wikitext text/x-wiki [[File:Viscount Mandeville Vanity Fair 1882-04-22.jpg|alt=Old drawing for a magazine of an aristocratic man in a formal suit|thumb|"Kim," Viscount Mandeville, 1882]] == Overview == The Mandeville who was in a relationship with music-hall and vaudeville star Bessie Bellwood was Kim (George Victor Drogo) Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892). He was Viscount Mandeville from 18 August 1855 (when his grandfather died) to 22 March 1890 (when his father died).<ref name=":1" /> When he succeeded to the dukedom in 1890, he ended his relationship with Bellwood. His father was the 7th Duke and his mother was [[Social Victorians/People/Louisa Montagu Cavendish|Louise Cavendish]], "double Duchess" of [[Social Victorians/People/Manchester|Manchester]] and then [[Social Victorians/People/Devonshire|Devonshire]]. Viscount Mandeville (right) was "Man of the Day" (#255) for ''Vanity Fair'', drawn by Leslie Ward ("Spy").<ref>"Viscount Mandeville Vanity Fair 1882-04-22.jpg." Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Viscount_Mandeville_Vanity_Fair_1882-04-22.jpg (retrieved 8 September 2026).</ref> == Also Known As == *Family name: Montagu *The title Viscount Mandeville is a courtesy title for the heir apparent and eldest son of the [[Social Victorians/People/Manchester |Duke of Manchester]]. This title is one of only 2 (for the heir apparent of a duke) that is below the rank of Earl (the other is the heir apparent of the Duke of Somerset). (Is Mandeville a toponym?) *Viscount Mandeville **"Kim" (George Victor Drogo) Montagu [8th Duke], (18 August 1855 – 22 March 1890) **[[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke (22 March 1890 – 18 August 1892) **Alexander George Francis Drogo Montagu (2 October 1902 – 9 February 1947)<ref>{{Cite journal|date=2020-10-15|title=Alexander Montagu, 10th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=Alexander_Montagu,_10th_Duke_of_Manchester&oldid=983660353|journal=Wikipedia|language=en}}</ref> *Viscountess Mandeville **Consuelo Iznaga y Clement Montagu [8th Duchess] (22 May 1876 – 22 March 1890) **Helena Zimmerman Montagu (14 November 1890 – 18 August 1892) *Lord Kimbolton (Baron Kimbolton of Kimbolton is the courtesy title for the heir presumptive, the heir apparent's eldest son, so the son of Viscount Mandeville<ref>{{Cite journal|date=2026-09-21|title=Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=Duke_of_Manchester&oldid=1375987697|journal=Wikipedia|language=en}}</ref>) ** [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke (1877 – 22 March 1890) == Acquaintances, Friends and Enemies == === "Kim" Montagu, 8th Duke, Lord Mandeville until 1890 === ==== Friends ==== * Edward Russell, 24th Baron de Clifford * Derrick Westenra, 5th Baron Rossmore * Francis Needham, 3rd Earl of Kilmorey === Consuelo Iznaga y Clement Montagu === ==== Friends ==== *Consuelo Iznaga y Clement Montagu was in the intimate circle of [[Social Victorians/People/Albert Edward, Prince of Wales |Albert Edward, Prince of Wales]]. ==Organizations== === "Kim" Montagu, 8th Duke, Lord Mandeville until 1890 === * Eton<ref name=":1" /> === Consuelo Iznaga y Clement Montagu === * Consuelo Iznaga y Clement Montagu was a member of the New York City social set called the Buccaneers when she was a teenager.<ref name=":0">{{Cite journal|date=2020-10-06|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=982192002|journal=Wikipedia|language=en}}</ref> Edith Wharton is said to have take details from the 8th Duke's marriage ''The Buccaneers''.<ref name=":1">{{Cite journal|date=2020-10-06|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=982188929|journal=Wikipedia|language=en}}</ref> == Timeline == '''1855 August 18''', the 6th Duke died. The families "placed in mourning":<blockquote>He is succeeded in the Dukedom by his eldest son, Viscount Mandeville, M.P. By the melancholy event the families of the Marquess and Marchioness of Tweeddale, the Duchess of Wellington, the Duke of Montrose, Lord and Lady Olivia Ossulston [?], Lady Eliza [Elisa?] Steele, the Duke of Richmond, the Countess of St Germans, Lady Sophia Cecil, and many others of rank, are placed in mourning.<ref>"Death of the Duke of Manchester." ''Edinburgh Evening Post and Scottish Standard'' 22 August 1855, Wednesday: 2 [of 4], Col. 5c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001177%2F18550822&page=2&article=025&stringtohighlight=mandeville+viscount. Print title ''Edinburgh Evening Post and Scottish Record''; n.p.</ref></blockquote>'''1855 October 11, Wednesday''', H. I. Cridland says in a "Notice,"<blockquote>I will hold Audit Days at Lyttelton, on Wednesday, 11th October, for the receipt of rents and other monies due to the 25th of September, on account of the Soames, Jackson, Ecclesiastical, and Lyttelton Trust Estates, the lands of Lord Mandeville, Lady Emily Bathurst, Lady Olivia Sparrow, Rev. J. P. Rhoades, Rev. Charles Hodge, Dr. Merrion C. T. Maunsell, George Hart, late William Draper, and J. E. Fitz Gerald, Esqrs. ... Tenants are requested to present their last receipt.<ref>"Notice." ''Lyttelton Times'' 6 October 1855, Saturday: 1 [of 10], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003346%2F18551006&page=1&article=004&stringtohighlight=mandeville+viscount. Newspaper title ''The Lyttelton Times'', n.p. </ref></blockquote>This notice was repeated in the ''Lyttelton Times'' on 29 December 1855. '''1855 October 20, Saturday''', the [[Social Victorians/People/Wilton|Earl and Countess of Wilton]] hosted a [[Social Victorians/Timeline/1850s#1855 October 20, Saturday|house party at Heaton Hall]], with a visit to a silk mill in town. [[Social Victorians/People/Louisa Montagu Cavendish|Luise, Duchess of Manchester]] was present with the Duke and their son Lord Mandeville (Kim, later 8th Duke), who was 2 years old. '''1876 May 22''', Consuelo Iznaga y Clement and "Kim" (George Victor Drogo, 8th Duke) Montagu married in Grace Church, New York City.<ref name=":0" /> === 1890s === '''1890 January 28, Tuesday''', Lady Manville attended what was supposed to be [[Social Victorians/People/Lily Langtry#1890 January 28, Tuesday|Lily Langtry's opening-night performance at the St. James's]] with the Prince of Wales and some of his circle. Also on this date, the ''Dundee Evening Telegraph'' reported that Lady Mandeville [[Social Victorians/Timeline/1890#1890 January 28, Tuesday|had bought 30 "Exploration" shares]], which funded exploration expeditions with the possibility of rights to minerals or resources or the development of monopolies for goods that were "discovered."<ref>"Notes — Mainly Personal." ''Dundee Evening Telegraph'' 28 January 1890, Tuesday: 2 [of 4], Col. 2b [of my 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000453%2F18900128&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''Evening Telegraph'', n.p.</ref> Her mother in law, [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess of Manchester]] also had bought 30 shares, both small amounts compared to some others. '''1890 January 29, Wednesday''', Consuelo Lady Mandeville was a guest at the house party around and the [[Social Victorians/Santurce Murrieta Wedding 1890-01-29|wedding of Miss Clarita de Murietta and the Duc de Santona]]. She had her eldest son with her, Lord Kimbolton (the 9th Duke of Manchester). Her mother in law, [[Social Victorians/People/Louisa Montagu Cavendish|Louise, Duchess of Manchester]] was also present. '''1890 February 6, Thursday''' '''(shortly before 6 February)''', Kim Lord Mandeville attended a [[Social Victorians/Timeline/1890#1890 February 6, Thursday|boxing match fought by the prize-fighter Pete Jackson]] at the New Pelican Club. '''1890 January 30, Thursday, through at least February 8, Saturday''', Kim Lord Mandeville was among the [[Social Victorians/Timeline/1890#1890 February 8, Saturday|recent arrivals at Brighton]] although apparently after the most notable had already left. Certainly he does not appear to have been at the party the next evening that the Prince of Wales attended. '''1890 February 9, Sunday''', the Prince of Wales came after one of Consuelo's dinner parties:<blockquote>Lady Mandeville, who has been giving a series of small dinners in her house in Cumberland Place, had a dinner on Sunday. The Prince of Wales came in after dinner to meet a small party.<ref>"Society Gossip. (From ''The World''.)." ''Leighton Buzzard Observer and Linslade Gazette'' 11 February 1890, Tuesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001091%2F18900211&page=6&article=106&stringtohighlight=mandeville+viscount. Print title: ''The Leighton Buzzard Observer'', p. 6. </ref></blockquote>'''1890 February 24, Monday''', Consuelo Lady Mandeville attended [[Social Victorians/Timeline/1890#24 February 1890, Monday|Lilly Langtry's performance as Rosalind in As You Like It at the St. James's Theatre]]. '''1890 February 25, Tuesday''', Consuelo Lady Mandeville attended a [[Social Victorians/Timeline/1890#25 February 1890, Tuesday|party hosted by Mrs. Adair]]. Viscount Mandeville is not listed among those who attended. '''1890 March 4, Wednesday''', Consuelo's brother Fernando Yznaga married New York "beauty and society leader" Miss Mabel Wright.<ref>"Personal." ''American Register'' 22 March 1890, Saturday: 5 [of 8], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003338%2F18900322&page=5&article=070&stringtohighlight=mandeville+viscount. Same print title and p.</ref> Presumably at least Consuelo was there? '''1890 March 10, Monday''', the Prince and Princess of Wales gave a [[Social Victorians/Timeline/1890#10 March 1890, Monday10 March 1890, Monday|dance at Marlborough House in honor of their wedding anniversary]]. The Viscountess Mandeville is listed among the guests invited, but the Viscount is not mentioned. '''1890 March 22, Saturday''', the Portsmouth ''Evening News'' expressed outrage about a man with a reputation like Kim's becoming 8th Duke of Manchester:<blockquote>While the House of Commons were debating the necessity for a House of Lords, the Duke of Manchester was on his death bed in Naples. There is more in the incident than appears at first sight, for Viscount Mandeville succeeds to the Dukedom, with the attendant rights of presentation to seven livings and the privilege of mutilating or rejecting Bills demanded by the people and passed by their representatives in the Commons.<p> It will hardly be urged by the moat zealous advocate of the Upper Chamber, as at present constituted, that the new Duke of Manchester, ''nee'' Viscount Mandeville, will bring much dignity, wisdom, or weight to the House of irresponsible legislators. This nobleman has had a merry time. The one sensible thing he did was to marry a charming and accomplished American lady, who, by consenting to share the name of Mandeville has kept its head above water. Viscount Mandeville himself has a weakness for bankruptcy proceedings, rows with cabmen, which music hall ladies come forward to take his part, and the class of events generally which do not approve themselves to the minds of the ordinary Briton. It is possible that Lord Mandeville may now discover that he has sown his wild oats, and may settle down with his beautiful and popular wife to the enjoyment of his  25,000 acres of land and £38,000 a-year. We have not to do with what Lord Mandeville may be. Our simple duty is to point out that it is an outrage on common decency and common sense that a man bearing the reputation of Lord Mandeville should have the right to step into the House of Lords whenever a vote is needed to swell the prejudice or sustain the selfishness that find among the Peers their natural home.<ref>"To-Day." ''Portsmouth Evening News'' 22 March 1890, Saturday: 2 [of 4], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000290%2F18900322&page=2&article=001&stringtohighlight=mandeville+viscount. Print title: ''The Evening News'', n.p.</ref></blockquote>The ''Birmingham Daily Mail'' had a similar opinion, though perhaps slightly more optimistic:<blockquote>The Duke of Manchester, who expired last night at Naples, cannot be said to have been much of a public man. He led a quiet, retired life, and the announcement of his death would, if Lord  Mandeville were not heir to the title, arouse but little interest. Lord Mandeville is, unfortunately for himself, more widely known than his father was. He developed tastes for gambling, with disastrous financial results, as the details of his bankruptcy made public last year, and more recently, his name has been publicly associated with those of prizefighters and music-hall singers. Altogether, he can hardly be termed a credit to the aristocracy of England, but the higher position to which he will now attain may cause him to alter his ways and try redeem the errors of the past.<ref>"Notes and News." ''Birmingham Mail'' 22 March 1890, Saturday: 2 [of 4], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000644%2F18900322&page=2&article=027&stringtohighlight=mandeville+viscount. Print title: ''Birmingham Daily Mail'', n.p.</ref></blockquote>On the same day, Consuelo's brother and new wife were reported to apparently be immigrating to the UK:<blockquote>Among the immigrants, says a London lady correspondent, is a New York belle who has just been married to Mr. Ferdinand Yznaga, brother to Lady Mandeville and of Lady Lister-Kaye. Young Mrs. Yznaga will enter London society under the wing of her sister-in-law, Lady Mandeville, and is expected to make as great a furore as did Mrs. Langtry a few seasons ago. Mr. Yznaga married a sister-in-law of Mr. Vanderbilt, who divorced him two years ago.<ref>"An American Belle." ''Huddersfield and Holmfirth Examiner'' 22 March 1890, Saturday: 6 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0003869%2F18900322&page=6&article=006&stringtohighlight=mandeville+viscount. Print title: ''The Huddersfield Examiner'', p. 6.</ref></blockquote>'''1890 November 14''', William Angus Drogo Montagu (9th Duke) and Helena Zimmerman married secretly, in London. '''1897 July 2''', the Duke and Duchess of Devonshire ( [[Social Victorians/People/Louisa Montagu Cavendish|Luise Friederike Auguste Gräfin von Alten Montagu Cavendish]] and [[Social Victorians/People/Spencer Compton Cavendish|Spencer Cavendish]]) hosted the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Lord Charles Montagu, likely Charles William Augustus Montagu, attended, as did [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]]. == Demographics == ===Nationality=== *Consuelo Iznaga y Clement was Cuban-American. *Helena Zimmerman was American. *George, William and Alexander Montagu were British. === Residences === * 1890: "The family estates — 27,312 acres, with a nominal rent-roll of £40,000 — were chiefly situated in Huntingdonshire and Armagh."<ref>"Men and Things." (London) ''Echo'' 22 March 1890, Saturday: 1 [of 4], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004596%2F18900322&page=1&article=010&stringtohighlight=mandeville+viscount. Same print title, n.p.</ref> * In 1890 at the death of the 7th Duke, Kimbolton Castle: <blockquote>The Duke of Manchester... He was an old-fashioned Tory, and a man of simple tastes, whose chief interests were military and agricultural rather than political. Kimbolton Castle is a magnificent place, and the Duke's horses and troop of yeomanry were always the admiration of the Prince of Wales, the Marquis of Hartington, and many other visitors. Lord Hartington, it may be mentioned, was at Kimbolton when the news came of the  breaking out of the war between France and Germany [1871], he being a member of the Cabinet. The Duke had travelled much as a young man, and was familiar with the Continent, as also with South Africa, of which he was very fond. There is a private chapel at Kimbolton Castle, connected with which is a very extensive library containing some theological and historical volumes of great value. Notwithstanding his somewhat bluff manner the Duke was liked by the tenantry and poor from the kindly interest he took in their concerns. Even the poachers had a good word for his Grace. The feature, however, of the Duke's career was the marked contrast between his pursuits and those of the Duchess. The Duke loved every blade of grass, every tree and flower on his ancestral domain, and cared nothing for London life in its modern developments. Lord Mandeville [the 8th Duke] succeeds to the title, but the estates are heavily mortgaged.<ref>"Death of the Duke of Manchester." ''Pall Mall Gazette'' 22 March 1890, Saturday: 6 [of 8], Col. 2b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000098%2F18900322&page=6&article=014&stringtohighlight=mandeville+viscount. Same print title and p. </ref></blockquote> == Family == *Consuelo (María Consuelo) Iznaga y Clement Montagu, Duchess of Manchester (1853 – 20 November 1909), née Doña María Consuelo Iznaga y Clement<ref name=":0" /> *"Kim" (George Victor Drogo) Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892)<ref name=":1" /> #William Angus Drogo Montagu, 9th Duke of Manchester (3 March 1877 – 9 February 1947) #Jacqueline Mary Alva Montagu (27 November 1879 – 15 March 1895). #Alice Eleanor Louise Montagu (27 November 1879 – 10 January 1900) *[[Social Victorians/People/William Angus Drogo Montagu|William Angus Drogo Montagu]], 9th Duke of Manchester (3 March 1877 – 9 February 1947)<ref>{{Cite journal|date=2020-10-08|title=William Montagu, 9th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_9th_Duke_of_Manchester&oldid=982513036|journal=Wikipedia|language=en}}</ref> *Helena Zimmerman Montagu (25 September 1878 – 15 December 1971)<ref>{{Cite journal|date=2020-10-08|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=982513630|journal=Wikipedia|language=en}}</ref> #Mary Alice Montagu Gregory (26 October 1901 – 9 October 1962) #Alexander George Francis Drogo Montagu, 10th Duke of Manchester (2 October 1902 – 23 November 1977) #Edward Eugene Fernando Montagu (26 July 1906 – 4 May 1954/1956) #Ellen Millicent Louise Montagu Hofer Shairp (5 January 1908 – 2 August 1948) === Relations === The three Iznaga y Clement sisters were prominent in London "Society": *Consuelo (María Consuelo) Iznaga Montagu, Duchess of Manchester (1853 – 20 November 1909) *Emilie Iznaga y Clement () *Natividad (Natica) Iznaga y Clement Lister-Kaye () == Questions and Notes == *In a ''British Newspaper Archive'' search for name=mandeville keyword=#viscount for 18 August 1855–22 March 1890 (September 2026), first did "Publication date oldest first," but Kim was 2 years old when his grandfather died and his father acceded to the dukedom, so on p. 57 of the search results (the page containing 1 January 1856) I switched to "Publication date newest first." Decided it would be more fruitful to work backwards. == Papers, Memoirs and Biographies == == Footnotes == {{reflist}} t29nt5lkqn7c1xw2l0fqy2einyemvb8 0 264724 2835525 2834160 2026-10-02T16:29:31Z Scogdill 1331941 2835525 wikitext text/x-wiki {{Short description|Dress worn by Queen Victoria at her wedding to Prince Albert in 1840}} = Sandbox = Page to draft revisions for Wikipedia articles. For Gwladys Robinson, see Gwladys Lowther Robinson, [[Social Victorians/People/Ripon|Marchioness of Ripon]] and, earlier, [[Social Victorians/People/Lowther|Countess of Lonsdale]] ==References== {{reflist|2}} [[Category:1840 works]] [[Category:Royal wedding dresses|Victoria Queen]] [[Category:1840s fashion]] [[Category:British royal attire]] [[Category:Dresses in the Royal Collection of the United Kingdom|Victoria, Wedding]] [[Category:Diamond Jubilee of Queen Victoria]] = Victorian fashion = ==Women's fashion== == Men's fashion == (Two intros: one to Victorian Fashion, giving background for both women's and men's fashions; two to the decades for men's fashion) Among the influences on men's styles: * differentiation of genders (domestic ideology, ideology of the spheres) * the "Great Male Renunciation," the French Revolution and English tailoring * industrial revolution: mass production of fabrics and articles of clothing, especially for working-class men * sports and country clothing * Albert Edward, Prince of Wales, later King Edward VII, influence went beyond England; tailors led by him led fashion for the upper classes * then later, cult of youth * military uniforms * comfort and convenience * evolving definitions of gender, both femininity and masculinity evolved as concepts. The changing definitions of masculinity affected men's clothing. For the growing and rising middle classes, gender roles became more and more rigid around the concept of the separate spheres, the public for men and the private sphere for women. Cultural sources for men's clothing and men's fashion: industrial revolution; expanding middle class; technological advancements; an association in the culture between outward appearance and inward nature; French Revolution; Albert Edward, Prince of Wales Citation for Victorian Hell: <ref name=":29">{{Cite web|url=https://victorianhell.substack.com/p/victorian-mens-fashion-history-and|title=Victorian Men’s Fashion History and Clothing Guide|last="It's Monty, Actually"|date=31 October 2025|website=Victorian Hell|access-date=7 September 2026}}</ref> === Frock coats === * An unidentified man wearing a double breasted frock coat - DPLA - 555cd4bb700ec3a18afbdf5c6638fe53 (page 1).jpg * Full-length portrait of a man, dress in frock coat and top hat with sash and pistol at his waist, seated on a crate LCCN2006689590.tif * Apostol Mănescu, politician, wearing a frock coat, butterfly tie, black waistcoat with button chain and light coloured pants, drainpipe type. Circa 1866. Photo by Franz Duschek.jpg (1866) * Oscar Wilde 3.jpg * The spring book of B. Kuppenheimer & Co. - illustrating a new century of clothing-radically different from the old-and incidentally depicting the more striking chances in men's wear during the (IA cu31924032172086).pdf * Clothes dictionary; (IA clothesdictionar00bake).pdf * FrockCoat.jpg (1870s) * Hint's about men's dress, right principles economically applied (IA hintsaboutmensdr00barr).pdf (book) * Men in frock coats, capes and top hats look at pictures pegg Wellcome V0039491.jpg (1813, but note the full coat) * Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg (used; 1850–1860s?, derby hats) * Full-length portrait of two young men wearing frock coats with sashes and pistols at their waists LCCN2006689564.tif * Fashion- the power that influences the world (IA fashionpowerthat00foxg).pdf (1871) * How to behave and how to amuse. A handy manual of etiquette and parlor games (IA howtobehavehowto00sand).pdf (1895; Men's Dress, p. 23) * The etiquette of men's dress (IA etiquetteofmensd00cluc).pdf (1888; search for "frock" — '''cutaway vs Prince Albert''') *S.W. Silver & Co.'s handbook for transatlantic travellers .. (IA swsilvercoshandb00silv).pdf (1883; has pages where ppl can write "Introductions" and "Engagements," but this is not a conduct book, it's sort of a guide to London) *Gazette of Fashion, October 1864, RP-P-2009-2673.jpg (1864; used; men + 1 boy; note trousers; Edward Minister & Son, Regent Street) *Gazette of Fashion, February 1865, RP-P-2009-2675.jpg (1865; used; men + 1 boy + 1 woman in a crinoline cage; Edward Minister & Son, Regent Street) *George Augustus Henry Sala.jpg (Mathew Brady, 1855–1865) *W Curtis Noyes.jpg (Mathew Brady, 1855–1865; use for coats, no trousers) *Photo of Prince Louis of Orléans, Prince of Condé.png (French, 1861; lighter version available) *Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg (Swedish, 1847) *[https://upload.wikimedia.org/wikipedia/commons/f/f6/How_we_go_to_those_who_cannot_come_to_us.._%28IA_howwegotothosewh00roge%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original How we go to those who cannot come to us.. (IA howwegotothosewh00roge).pdf] (Roger, Peet & Co.; American, pp. 11, clerical frock; 7 "Prince Alberts" double-breasted frocks; 23 shirts as underwear) * ['''new content'''] Although women followed Paris fashion in Europe and America, London tailors led the way for men’s fashion. The Industrial Revolution actually impacted men’s fashion more than women’s.  Sewing machines in factories began mass producing cheap clothing for the largest group in the population, working class men. Ready-made clothing affected the social structure, the economy, and gender differentiation. Paper patterns, introduced in fashion magazines that targeted women, along with domestic sewing machines had more impact on women’s fashion than men’s. Haute couture was still bespoke (being individually constructed by tailors or dressmakers or fashion houses like Maison Worth). === Industrial Revolution and Technological Advancements === * the railroad * mass production of fabrics for working-class men's clothing ** Jacquard looms ** * sewing machines * aniline dyes, including black === Expanding Middle Class === * gender roles getting more clearly defined and rigid (referring back to page overview) === French Revolution === Sarah Gharmallah Alzahrani and Safia Abdelaziz Saroukh (https://www.researchgate.net/profile/Safia-Saroukh/publication/385099194_The_Semiotic_Dimension_of_Men's_Fashion_in_Modern_Eras/links/671686fbd796f96b8ec4f90e/The-Semiotic-Dimension-of-Mens-Fashion-in-Modern-Eras.pdf):<blockquote>The 19th century started with a fashion landscape that was changing dramatically and rapidly from the styles of a generation earlier. The French Revolution brought fashions that had been emerging since the 1780s to the forefront. Neoclassicism now defined fashion as both men and women taking inspiration from classical antiquity. For women, the high-waisted silhouette in lightweight muslin was the dominant style, while fashionable men looked to the tailors of Britain for a new, refined look [17]. [17]  Franklin, H. (Aug 18, 2020). Published on Jun 25, 2020, Retrieved: <nowiki>https://fashionhistory.fitnyc.edu/1800-1809/</nowiki> 11/11/2023. Edited. ...</blockquote>Brent Shannon:<blockquote>"Costume," wrote Max Beerbohm in 1896, "enables us to classify any 'professional man' at a glance, be he lawyer, leech or who not" (24–25). A man's profession and class were read by his jacket, his hat, what he rode in, and how he carried himself. "Perhaps there is a tendency among Englishmen to judge a man too much by the shape of his hat or the kind of collar he wears," conduct author John Wanamaker confessed; "But one must remember that in England if you ''wear'' the wrong thing, you will probably ''do'' the wrong thing, and generally ''be'' the wrong thing" (1).¹¹ Such assertions were predicated on the powerful Victorian conviction that outward appearance reflected inner qualities.<ref>{{Cite book|title=The Cut of His Coat: Men, Dress, and Consumer Culture in Britain, 1860–1914|last=Shannon|first=Brent Alan|publisher=Ohio University Press|year=2006|location=Athens, Ohio}}</ref> (148)</blockquote> == Victorian Men's Fashion == David Kutcha traces the origin of the 3-piece men's suit to King Charles II of England, who may be said to be wearing a coat, pants and a waistcoat, but otherwise is not wearing a "suit" we — or the Victorians — would recognize as such: <blockquote>... since 1666, male gentility has been associated with modesty and plainness in dress. Eschewing fashion as an increasingly feminized realm Charles II's vest inaugurated a new and essentially modern era of masculine aesthetics, one that reversed a long-held association between elaborate display and high social status. Manly thrift now displayed elite status.<ref>{{Cite book|title=The Three-Piece Suit and Modern Masculinity, England 1550–1850|last=Kutcha|first=David|publisher=University of California Press|year=2002|location=Berkeley and Los Angeles}}</ref> (2)</blockquote>So it's really Cromwell who should get credit?? The French Revolution caused both men and women to change stylistic direction radically. For one generation, stylish women wore simple white muslin dresses without foundation garments underneath. The changes in men's fashion was more lasting: men turned away from the ornamentation and ruffles of the ''ancien régime'', choosing instead understated and subtle tailoring to indicate good taste and wealth. Beau Brummell: "Men began to aspire to fine cutting, tailoring, and perfect fit in their clothes, flawless grooming and manners in themselves. The Englishman George Bryan (Beau) Brummell deserves much credit for the ideal of meticulous masculine appearance."<ref name=":25">{{Cite book|title=The History of Costume: From Ancient Mesopotamia Through the Twentieth Century|last=Payne|first=Blanche|last2=Winakor|first2=Geitel|last3=Farrell-Beck|first3=Jane|publisher=Addison-Wesley Longman|year=1992|isbn=0-06-047141-7|edition=2nd|location=New York, New York}}</ref> (458) "From 1796 to 1816, the Beau set the pattern for cleanliness and liberal use of starch." (459) '''Victorian gender ideologies''' relegated men to the "public sphere" and women to the "domestic sphere." (??) More than that “the cult of youth . . . the secularization of sport and the influence of modern warfare in generating men’s fashions, the tendency to prefer modern ideas of comfort and convenience, and the transformation of fashion knowledge and fashion urbanism from print to hyperreality”<ref name=":26">McNeil, Peter. "Men's Fashion: 1800–2022." Chapter 22. ''The Routledge History of Fashion and Dress, 1800 to the Present''. Routledge, 2024. https://opus.lib.uts.edu.au/bitstream/10453/182707/2/Men%27s%20Fashion%20200822_24_12_20_09_29_44.pdf DOI: 10.4324/9780429295607-27.</ref> (McNeil 1 or 411) changed the perceptions of masculinity and femininity. The growing middle class involved among other things more and more jobs and careers for young men as clerks, office workers, they were junior, subordinate, and they were commuting on the railroads from suburbs. Carolyn Kirby:<blockquote>In western Europe the fashion for plain dark suits coincided with the rise of the affluent middle-classes in a world where the pace of industrialisation and the globalisation of trade was accelerating as never before. The sharp, dark business suit became the last word in male power-dressing. And so it remains to this day.<ref name=":27">{{Cite web|url=https://historiamag.com/invent-masculine-fashion/|title=The invention of masculine fashion|last=Kirby|first=Carolyn|date=3 December 2025|website=Historia: Magazine of the Historical Writers' Association|access-date=25 August 2026}}</ref></blockquote>Albert Edward, Prince of Wales was very concerned with fashion and authoritative about it, with a very specific eye to small details. He was increasingly influential in men's fashion from the last third of the 19th and into the 20th century, and many noticed what he did and copied him.<blockquote>When that great lover of pleasure, Edward VII, visited Marienbad incognito as the Duke of Lancaster, he was followed by tailors from Paris, Budapest, Vienna, and Berlin who photographed him and took notes about his clothes. Edward VII introduced many [423–424] novelties into men’s fashion. ... He is also credited with making fashionable the creased trouser in 1909 (his groom dried them with a board weight after heavy rain, resulting in the line), turned-up cuff trouser (after hitching his trouser bottoms at a dirty racing track) and, as his girth grew, undoing the bottom button of his waistcoat."<ref name=":26" /> (424)</blockquote>Cowles<blockquote>The Prince was not just ‘a swell’. In the jargon of the day he was ‘a heavy swell’, and apparently there was a world of difference between the two terms. A swell was a rich young aristocrat who lived in extreme comfort; but a heavy swell added showmanship to the comfort and lived in a stylish luxury that even the French were obliged to envy. And of course the heavy swell was the acme of sartorial elegance. The Prince did not mind changing his dress half a dozen times a day. He loved clothes, and since whatever he chose to wear became the prevailing fashion overnight, he soon was regarded as an expert on the subject. His tailor-in-chief made a fortune. For many years he patronised a Mr. Poole. He discovered this gentleman by accident. He went to the theatre one night to see a well-known actor by the name of Fecher playing ‘Robert Macaire’. As an impecunious adventurer [129–130] Fechter was obliged to wear a coat that was torn and dirty, but Bertie’s expert eye noticed the elegant cut. At the end of the performance he asked Fechter for the name of his tailor, and Mr. Poole’s future was assured. The Prince had so many clothes he could never travel with less than two valets; and two more valets were left at home cleaning, brushing and pressing his vast wardrobe. There were suits and coats for every variation of every climate the world over. There were over a hundred pieces of headgear; and since Bertie was an honorary admiral and an honorary general of most of the countries of Europe, there was an entire room devoted to uniforms, sashes, epaulettes, belts, buckles, swords, feathers and other regalia. ... Men’s clothes became of such importance that new [130–131] shops sprang up like mushrooms in Savile Row, Clifford Street and Bond Street. Most of the Prince’s innovations were inspired by comfort and convenience. He altered the cut of the evening dress waistcoat, he shortened the tails on the tail coat, he left his frock coat open (due to an increasing girth), he introduced the black homburg, and he attended race meetings, not in the frock coat hitherto ''de regueur'' but in tweeds. He tried having his trousers creased down the sides rather than the front and back, in order to hide his bandy legs, but this idea did not catch on, and he soon discarded it himself. But the prince was not the only arbiter of men’s fashions. The band of  "heavy swells" who followed his lead gave him plenty of competition. Lord Raglan and Lord Petersham invented coats which are still named after them. Lord Dupplin the dinner jacket and Lord Cardigan the button-up sweater. But Lord Hardwicke made the most spectacular contribution. Men’s silk hats were made of beaver which was left in its original rough, shaggy state. Lord Hardwicke polished his hat until he could see his face in it, and consequently was known as "Glossy Top". He is responsible for the top hat as we know it today.<ref name=":28">{{Cite book|title=Gay monarch, the life and pleasures of Edward VII|last=Cowles|first=Virginia|publisher=Harper|year=1956|location=New York, New York|archive-url=https://archive.org/details/gaymonarchlifepl0000cowl/}}</ref> (129–131)</blockquote>Poole "comfort and convenience" — trousers, sack coats, collars and cuffs without a shirt, wool rather than silks and satins; knee breeches constantly riding up men as leaders of fashion, in spite of our sense that men didn't do fashion in the way that women did, were more serious than that ==== Men’s Clothing 1830 -1900 ==== “Tailored wool garments , originally inspired by the clothing of English country gentry, formed the backbone of men’s wardrobes in the early nineteenth century.”<ref name=":25" /> (458) Men’s clothing began to change from styles that were colorful and flamboyant in the eighteenth century to styles that emphasized cut over color. By the end of the century men’s clothing had settled into a kind of uniform with coats and trousers usually made of the same fabric, often with a vest or waistcoat of the same fabric. The uniform included a shirt with a stiff collar, a necktie and very little jewelry. It took the entire nineteenth century to change to a rather static style that is still worn today. (this is middle-class men for day wear?) ==== Silhouette ==== The silhouette for men sometimes followed the same line as women’s clothing on the top half of their bodies.. For example, as women’s sleeves grew into the huge leg-of-mutton fullness, the shoulders of men’s coats also had added fabric to make a pouf at the shoulder, thus emphasizing a broad shouldered look across the front. Men’s silhouettes changed subtly without the exaggerated fullness of women’s skirts or the raising and lowering of the neckline. Once men began to wear trousers, they did not go back to formal knee breeches (except in European courts or for court wear for Victoria). The line of men’s garments changed in smaller increments than women’s. Coats and trousers might fit tighter in some decades and looser in others. They were slim, then boxy, flared, then straight and evolved into the contemporary line we see on men today. ==== Coats and Shirts ==== Several styles of coats were available to fashion-conscious aristocratic and upper-middle-class men. Men’s suits in this period were tailor made, fitted to one specific man. The factories, however, knew that the vast majority of men were working class and began to manufacture and sell  ready-made shirts, coats and trousers. More clothes were available for working men and they cost less. The cheaper work clothes enabled upper-working-class men to own more than one set of clothes and to have more formal clothes for church or other events. Ready-made shirts were also available in generic sizes, small, medium and large. Generic sizing speeded up the mass production process and, since little of the shirts could be seen, they did not need custom fitting. [[File:Frock coat MET CI38.23.39 F.jpeg|alt=Old-fashioned coat hanging on a mannequin|thumb|Frock coat, 1840, American]] Frock coats were introduced in the 1820s and worn throughout the nineteenth century with only minor changes to the line or fit. Sleeves were full at the top and waists were tight, creating an hourglass form. Initially frock coats were tightly fitted and closed with buttons. The more tailored frock coats had a waist seam and a flared skirt almost to the knees. The rate of fashion change notable in women's fashion was true for men's clothing as well. The frock coat persisted for a century, but it was in a state of constant (but subtle) evolution, with collars and lapels, coat length, the waist seam, the fullness of the coat itself as well as the sleeve, color, patterns like stripes or plaids. Men around the world and especially in the British Empire wore frock coats: images and descriptions exist of men wearing frock coats in India<ref>Woodyatt, Nigel Gresley. ''Under Ten Viceroys; The Reminiscences of a Gurkha''. London: H. Jenkins Limited, 1922. Pp. 31, 107, 1774.</ref>, Iran (Persia at that time)<ref>Sparroy, Wilfrid. ''Persian children of the royal family; the narrative of an English tutor at the court of H. I. H. Zillus-Sultán, G.C.S.I''. London and New York, J. Lane, 1902. Pp. 11, 29, 30.[https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf?utm_source=commons.wikimedia.org&utm_campaign=index&utm_content=original https://upload.wikimedia.org/wikipedia/commons/d/d1/Persian_children_of_the_royal_family%3B_the_narrative_of_an_English_tutor_at_the_court_of_H._I._H._Zillus-Sult%C3%A1n%2C_G._C._S._I_%28IA_persianchildreno00sparrich%29.pdf].</ref>, Japan<ref>Watson, W. Petrie. ''Japan, Aspects and Destinies''. New York, Dutton: 1904. P. 7, 19, 31.</ref>, Egypt and Turkey<ref>Butler, Alfred Joshua. ''Court Life in Egypt''. London: Chapman and Hall, 1887. P. 7.</ref>, .... Almost all of the coats Victorian men wore could be single or double breasted, with a couple of exceptions like morning coats and Norfolk jackets. On more formal occasions upper-class men wore a cutaway morning coat with light trousers during the daytime, and a dark tail coat and trousers in the evening. Shirts and collars separated: "by 1827 detachable collars became available."<ref name=":25" /> (477) Shirts were considered more like underwear early in the century but by the end of the 1800s, they were becoming the outside garments popular in the twentieth and twenty-first centuries. Neck treatments (and shirt cuffs) ==== Overcoats ==== Overcoats became a significant fashion element and many styles were seen throughout the century. <ref name=":24" />Kay, Fiona and Neil R. Storey, ''Victorian Fashions for Women'', Pen & Sword History, Yorkshire - Philadelphia, 2022 ==== Trousers ==== Breeches: Men’s bifurcated garment that ended just below the knee Trousers: men’s bifurcated garment that went down to the ankles or foot The most sweeping change in men’s fashion before 1820 was in the length of their pants. Trousers that went from waist to foot were so much more comfortable and popular than knee breeches that they replaced what had dominated men’s wear since the sixteenth century. And trousers have been worn for over two centuries now. They were originally tailored but began to be mass produced. Although the aristocracy were required to wear knee breeches at court, they wore trousers more than they wore breeches, and they could afford to have their suits and shirts hand sewn and individually. Working class men had to contend with small, medium, and large from mass production. This bifurcated garment was modified in specific places, like the length, the fullness over the hips or the width of the leg. In summer trousers were lighter in color — "grayish blue, aqua and pearl gray." (Payne 2, 505) In winter trousers mixed darker colors like "steel gray and brown, black and brown, and black and green." (Payne 505) More on summer colors, winter colors? [Find quote about QV's men's court dress and the men being so cold] ==== Waistcoats ==== Victorian men put their creativity, color and specialty fabrics into their waistcoats. At the beginning of the century men's waistcoats were made from specialty fabrics, like velvet or brocade or a patterned silk, '''sometimes two at the same time''': "During the 1820s and 1830s, men sometimes wore two waistcoats, in combinations like white velvet over rose-and-gold brocade."<ref name=":25" /> (474) Long after coats and trousers were made from the same dark wool, waistcoats continued to be colorful, making them the focus of the ensemble. By the end of the 19th century waistcoats made from the same fabric as the trousers and coat dominated men’s fashion. 1880s: "The term vest now applied to all but low-necked formal styles, which were still called waistcoats." (Payne 2 528) ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hats]] were worn by the working class. 1830s Men wore [[top hats]], with wide brims in sunny weather. 1830s - 1900 Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular.[[File:Men's fashion silhouette of 1837.jpg|thumb|Men's coats, 1837]] [[File:Lev Wittgenstein by F. Kruger.JPG|thumb|Lev Wittgenstein by F. Kruger, 1836]] === 1830s and 1840s === Men's fashions in the 19th century have been less researched than women's fashions, in part because of fewer artifacts of what men wore were saved in museums for study. Photography, the first century that we have actual images (rather than artists' interpretations) of what people wore.[[File:Man's Frock Coat LACMA M.2007.211.61 (1 of 4).jpg|alt=Male mannequin wearing a light-colored thigh-length coat similar to a suit jacket and a top hat|thumb|Men's frock coat, c. 1845]] [[File:Stockholms mode-journal- Tidskrift för den eleganta werlden 1847, illustration nr 2.jpg|thumb|Stockholm's fashion magazine, 1847]] ==== Silhouette ==== Some men wore corsets, so the waist was nipped in and the skirt of the frock coat flared out. The sleeves were slightly puffed at the shoulders. The trouser legs were straight with a strap under the instep. The figure on the left in the fashion plate from ''Modes de Paris Journal des Tailleurs'' (right) illustrates this silhouette very clearly. ==== Coats and Shirts ==== Frock coats became fashionable for day wear in the 1820s<ref name=":25" /> (474) and were worn throughout the century. The main changes in frock coats in these two decades were in the colors and the neck treatment (collar and lapels). Initially frock coats were brightly colored, like "plum, mulberry, brown, bright blue, and bottle green,"<ref name=":25" /> (474) although the truly bright, saturated colors made possible by synthetic dyes were a couple of decades away still. Middle- and upper-class men limited their use of tailcoats to evening wear and more formal occasions. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[cravats]] or neck ties. Before shirts were mass produced, they were tailored for upper and wealthy middle-class men. ==== Trousers ==== "Trousers of the 1830s fit quite snugly ..., but in the 1840s they became looser."<ref name=":25" /> (477) Knee breeches still existed, but only for court wear, riding and country wear (in the form of knickerbockers). ==== Waistcoats ==== waistcoat; vest “Waistcoats [or vests] remained colorful, decorative items in men’s wardrobes after the rest of their costumes had become plain.” (Payne 2 460-462) They could be single- or double-breasted, with shawl or notched collars, and might be finished in double points at the lowered waist. Collars gradually lowered over the course of the two decades. Often the colorful vests were decorated with fur or velvet collars. ==== Original Text ==== During the [[1840s in fashion|1840s]], men wore tight-fitting, calf length [[frock coat]]s and a [[waistcoat]] or vest. Sleeves were full at the top and waists were tight, creating an hourglass form. For more formal occasions, a cutaway morning coat was worn with light trousers during the daytime, and a dark tail coat and trousers was worn in the evening. Shirts were made of linen or cotton with low collars, occasionally turned down, and were worn with wide [[Cravat (early)|cravat]]s or neck ties. Men wore [[top hat]]s, with wide brims in sunny weather. [[File:Four young men wearing frock coats - DPLA - 4706a21c9267c45034d51c8fc5309251 (page 1).jpg|thumb|American, 1850–60s?, derby hats]] === 1850s === The 1850s revealed several industrial inventions that significantly impacted clothing. First, the Industrial Revolution began (and still does today) with machines to weave fabric and machines to sew fabrics. Manufacturers quickly learned that the majority of the population was working class. Theirs were the simplest, least trimmed garments made in the nineteenth century. Second, mass production of shirts, trousers and coats changed how most people shopped for clothes. The aristocracy and upper middle class folks could still have their clothes made to specifically fit their bodies. Manufacturers knew that they could not mass produce clothing for each individual’s body. They invented generic sizing in small, medium, and large. They could, therefore, produce many more items of clothing with generic sizing. Third, Isaac Singer patented and began to manufacture home or domestic sewing machines (treadle machines) enabling more form-fitted '''dresses''' to be constructed for an individual’s specific shape and size. The fourth item to impact fashion and clothing was the invention of the paper pattern. These were introduced in fashion magazines targeting women. Almost all women were taught to sew by hand, whether they were aristocrats or working class. Sewing machines enormously speeded up the '''process of constructing from hours to minutes'''. Finally, aniline dyes made colors brighter and longer lasting. Prints and stripes, checks and plaids, fabrics had much more vibrant color with aniline dyes. The Industrial Revolution impacted the working class and changed their lives the most. children's clothes? Women made men's clothing too According to Judith Flanders, working-class men's clothes could be colorful. For example drivers of hansom cabs () dressed more up-scale, and drivers of hackneys () did not.<blockquote>While hackney drivers were also considered to be stereotypically shabby, hansom-cab drivers were generally represented as smartly dressed. A print in 1850 showed a driver in a snappy brown coat instead of the coachman’s heavy multiple-caped outfit, pale green striped trousers, short boots and top hat, the [167–168] reins held daintily in his gloved hands. Both cab and coach drivers wore top hats, but cabbies of a sporting bent later switched to bowlers, and in summer donned bright checked outfits.<ref name=":23">{{Cite book|title=The Victorian City: Everyday Life in Dickens' London|last=Flanders|first=Judith|publisher=Thomas Dunne Books|year=2012|location=New York, New York}}</ref> (167–168 [of 972])</blockquote> '''Silhouette''' A variety of silhouettes was seen in the 1850s. Trousers were long and slim with a strap under the instep of the shoe. '''Coats and Shirts''' “Although the frock coat was worn throughout the century, it lost its dominant position in the 1850s, when the morning coat began to replace it.”<ref name=":29" /> A jacket shorter than the frock coat or morning coat that had visible pockets began to be worn in about 1850. By the 1860s, it had become the most popular casual coat. Cutaway tail coats were still reserved for formal occasions. All of these coats could be single or double breasted. During the [[1850s in fashion|1850s]], men started wearing shirts with high upstanding or turnover [[collar (clothing)|collars]] and [[necktie#Four-in-hand|four-in-hand necktie]]s tied in a bow, or tied in a knot with the pointed ends sticking out like "wings." '''Trousers''' Trousers had fly fronts and breeches were used for formal functions and knickers (short pants fuller cut than breeches) for horseback riding. Men's choice to wear trousers is an instance of the importance of comfort and convenience outweighing style and presentation, a value that increased in importance through the 20th century. '''Waistcoats''' Waistcoats remained the one garment that could be brightly colored and copiously trimmed on exotic fabrics. Waistcoats show that men were concerned with decorative elements in their ensemble. ==== Notes & Quotes ==== * transition from frock coats to ditto suits, 1850s (Payne, 463) ==== Hats ==== The upper-class continued to wear top hats, and [[bowler hat]]s were worn by the working class. Caps [[File:Gazette of Fashion, February 1865, RP-P-2009-2675.jpg|thumb|1865, Frock coats, etc. (Paris)]] [[File:Gazette of Fashion, October 1864, RP-P-2009-2673.jpg|thumb|1864, Frock coats, etc., Paris]] [[File:Broadway fashions, New York LCCN2003677917.jpg|thumb|Butterick, Spring-Summer 1866, Broadway Fashions]] === 1860s === [[File:Corporal frock coat 1862 med.jpg|thumb|Corporal frock coat 1862, 9-botton military]]The 1860s saw fewer inventions that influenced or changed the way people were acquiring clothes. But the changes begun by mass production continued with the working class. Little changed for the higher classes, except maybe paper patterns and sewing machines. '''Silhouette''' The different shape and size in clothing continued to provide slightly different silhouettes for men in the upper and middle classes. As women's clothes grew more complex in cut and style, men's clothes became simpler in cut and style. '''Coats and Shirts''' Individualized fittings continued for upper class and wealthy middle class clients. Working class as well as middle and upper class. Morning coats still led the way but frock coats were still in demand and worn in business and more casual situations. Leisure wear like knickers and ditto suits became more popular in the 1860s. Haute couture wanted every occasion or event to have special clothing. In some ways men dressed more alike (coat, trousers, waistcoat) and in some ways there was more variety in the cut of their coats and in the manufactured clothing they could buy. Many upper class men still had their shirts custom made. Frock coats were shortened to knee-length and were worn for business, while the mid-thigh length [[sack coat]] slowly displaced the frock coat for less-formal occasions, with the overall effect of a looser silhouette. The Prince of Wales: "For the countryside such as at Sandringham, he permitted an informal dress code. The Henry Poole ledger marked as 'HRH 1865' is for an evening coat without tails, the first 'dinner jacket.'"<ref name=":26" /> (424) Shirts '''Trousers''' Trousers were cut full and boxy during most of the 1850s and 1860s. The strap under the shoe disappeared and trouser legs reached the back of the heel in length. Knickers were still worn for hunting or riding and breeches were still ''de rigueur'' at court. '''Waistcoats''' Waistcoats remained decorative and worn with most of the coats that were popular. The young “dandies” wore the most colorful and decorated waistcoats. These “dandies” gave their highest priority to their appearance. They were the avant garde of haute couture. ==== Necks ==== In the [[1860s in fashion|1860s]], men started wearing wider neckties that were tied in a bow or looped into a loose knot and fastened with a stickpin. ==== Hats ==== Top hats briefly became the very tall "stovepipe" shape, but a variety of other hat shapes were popular.[[File:Mens Coats 1872 Fashion Plate.jpg|thumb|upright|Drawing of Victorian men 1870s]] [[File:Ashurst frock coat.jpg|thumb|Ashurst Frock coat]] [[File:Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat; Standing (fbc266c2feca43f6ad5b44e031635a8d).tif|alt=Old photograph of a man wearing a dark thigh-length coat similar to a suit coat with light-colored trousers|thumb|Grand Duke Alexis Alexandrovich of Russia Wearing a Frock Coat, c. 1872]] [[File:Caillebotte Hugot.jpg|thumb|Portrait of Paul Hugot, 1878]] === 1870s === To correct and prevent errors being made in court dress, in 1875 the Lord Chamberlain published ''Dress Worn by Gentleman at Her Majesty's Court'', "a summary of regulations for court uniform and dress."<ref>{{Cite book|url=https://www.google.com/books/edition/Dress_worn_by_Gentlemen_at_Her_Majesty_s/pvrbQCXq0MEC?hl=en|title=Dress worn by Gentlemen at Her Majesty's Court|last=Britain)|first=Victoria (Queen of Great|date=1875|language=en}}</ref> This is the kind of thing Bertie really cared about. Define court dress. '''Silhouette''' A portly, boxy silhouette was sought for most of the 1870s. Older men whose stomachs had spread outward and thus looked prosperous to the Victorians were the ideal for men’s high fashion. Sack coats and trousers were cut a little wider in the legs. '''Coats and Shirts''' There were minor changes to some of the coats Victorian men wore. Frock coats were shortened to the knee. Sack coats (or ditto coats — the forerunner of the suits worn by men in the western cultures of the twentieth and twenty-first centuries) were found to be so comfortable that most men of the upper and middle classes had a ditto suit. Men still wore morning coats and the aristocracy still wore cutaway coats with trousers to parties and balls but cutaway coats with knee breeches at court. Although ready-made shirts were available, the upper and middle classes still wore bespoke clothing hand-made, with better fabric and an excellent fit but stitched on machines. The suits and shirts made for the working class were of cheaper fabrics. Corduroy for trousers and muslin for shirts was often used in working class clothing.. '''Trousers''' Most men were wearing trousers  all of the time (except at court) and have worn them ever since. Trousers had button flies by the 1870s which most men found to be much more convenient than laces. Trousers did not “ride up” the leg in the way that knee breeches did when dancing or sitting. If a man did not have fashionable legs, they could be obscured behind trousers. '''Waistcoats''' ==== Original Text ==== During the [[1870s in fashion|1870s]], three-piece suits grew in popularity along with patterned fabrics for shirts. Neckties were the four-in-hand and, later, the [[Ascot tie]]s. A narrow ribbon tie was an alternative for tropical climates, especially in the Americas. Both frock coats and sack coats became shorter and more form fitting. Flat straw boaters were worn when boating. [[File:Elliott & Fry10a.jpg|thumb|William Gladstone, 1887, morning coat]] [[File:British politicians gambling at Ascot Wellcome V0050378.jpg|thumb|British Politicians Gambling at Ascot, 1884]] === 1880s === ==== Silhouette ==== The silhouette narrowed and straightened. That is, the line went straight down without the flare of skirts on the coats. This smooth line had few decorations. Men wore hardly any jewelry. They might have a boutonniere or a ring.  Suits were not  decorated. All coats were straighter and shorter. Depending on the time of day, the Prince of Wales might change his clothes (thus, his silhouette) up to six times per day,<ref name=":28" /> (129) and some aristocrats and upper-class men would follow his lead. ==== Coats and Shirts ==== Although the frock coat was worn throughout the nineteenth century, the single fabric sack suit was worn for casual wear and was beginning to be seen as a businessman’s suit.  Morning coats were still in use for some events as well as casual occasions. The modern tuxedo or Cowes jacket appeared and was worn for formal occasions. The tuxedo is still worn as formal dress today. Shirts were still custom made for the rich and the working class bought ready-made shirts. The shirt remained basically the same but it could handle the different neck treatments.  Even mass produced shirts could have buttons or button holes for detachable collars and cuffs. ==== Trousers ==== Trouser legs began to decrease in width as the coat became narrower and emphasized strong vertical lines. ==== Waistcoats ==== During the 1880s waistcoats were called vests and much of their bold trim and colorful fabric disappeared and vests were constructed from the same fabric as the coat and trousers. The business suit for men developed in the 1880s and was firmly established in the 1890s. ==== Original Text ==== During the [[1880s in fashion|1880s]], formal evening dress remained a dark tail coat and trousers with a dark waistcoat, a white bow tie, and a shirt with a winged collar. In mid-decade, the dinner jacket or [[tuxedo]], was used in more relaxed formal occasions. The [[Norfolk jacket]] and tweed or woolen breeches were used for rugged outdoor pursuits such as shooting. Knee-length topcoats, often with contrasting velvet or fur collars, and calf-length overcoats were worn in winter. Men's shoes had higher heels and a narrow toe. [[File:American fashions - The Jno. J. Mitchell Co. Litho., New York. LCCN2006676665.jpg|thumb|Fashion plate from the Jno. J. Mitchell Co., New York]] [[File:Alfred Milner Vanity Fair 1897-04-15.jpg|thumb|Alfred Milner, 1897, Leslie Ward ("Spy"), for Vanity Fair]] [[File:Nobl-08.jpg|thumb|Sherlock Holmes, 1892, Sidney Paget]] === 1890s === ==== Silhouette ==== The dominant silhouette for men became the sack suit in the 1890s and while changes in the sack have been subtle, it remains the dominant silhouette today. ==== Coats and Shirts ==== “Sacks became ''the'' favored style of suit coat,”(Payne2, p. 532) especially among the younger men. Frock coats were associated with older men along with the cutaway coat. There were many more choices for sports wear and leisure wear than ever before. Norfolk jackets were often worn while hunting. Most informal styles could be purchased ready made. The rich still had their clothing custom made by tailors but even working class men could purchase most ready-made versions of the coats, trousers, and vests. Shirts could be made of patterned fabric as well as solid colors and white.  Shirts were beginning to be acceptable as outer wear especially when worn as sporting clothing. ==== Trousers ==== Early in the 1890s, “slim, straight trousers were news.” (Payne 2, p. 532) Trousers could be plaid or checked as well as the same fabric as the coat and vest. ==== Waistcoats ==== Fancy fabrics and bold trim were not seen much as the more subdued vest dominated the fashion scene. ==== Original Text ==== Starting from the [[1890s in fashion|1890s]], the [[blazer]] was introduced, and was worn for sports, sailing, and other casual activities.<ref>{{cite web|last=Landow|first=George|url=http://www.victorianweb.org/art/costume/90s/2.html|title=Men's informal sporting dress, late 1880s and '90s}}</ref> Throughout much of the Victorian era most men wore fairly short hair. This was often accompanied by various forms of facial hair including moustaches, side-burns, and full beards. A clean-shaven face did not come back into fashion until the end of the 1880s and early 1890s.<ref>{{cite web|url=http://www.victorianweb.org/art/costume/nunn21.html|title=Victorian Men's Fashions, 1850–1900: Hair}}</ref> Distinguishing what men really wore from what was marketed to them in periodicals and advertisements is difficult, as reliable records do not exist.<ref name="shannon597">{{cite journal|last=Shannon|first=Brent|title=Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914|journal=Victorian Studies|year=2004|volume=46|issue=4|pages=597–630|doi=10.1353/vic.2005.0022}}</ref> === Notes === *Men's suits buttoned higher up than today (Payne, 467) * Norfolk jackets and sack suits (Payne, 471) * formal attire, tuxedos with tails, cutaways (Payne, 469) * Keith Middlemas (https://archive.org/details/storyoffiesta00huxf/page/200/mode/2up?q=fashion) *Brent Shannon. "Refashioning Men: Fashion, Masculinity, and the Cultivation of the Male Consumer in Britain, 1860–1914." Victorian Studies 46, no. 4 (Summer 2004): 597–630. == Hats and headwear == [[File:Ford.madox.brown.last.emma.study.jpg|thumb|''Emma Hill'' by [[Ford Madox Brown]] (1853), a woman wearing a later version of the [[poke bonnet]]]] [[File:Hoed,_objectnr_KA_1237.tif|left|thumb|Perched bonnet style of the early 1870s.]] Hats were crucial to a respectable appearance for both men and women. === Men's Hats === The top hat, for example, was standard formal wear for upper- and middle-class men.[Payne] According to Blanche Payne, "The high top hat, usually black or dark gray, had reached its characteristic shape by 1798 and dominated the entire nineteenth century." (457–58) Although top hats were the dominant hat in the 19th century, other hats became popular for working classes and lower income middle class. “The style of an individual’s hat varied, depending on fashion and their social position, as well as their profession or chosen activity..” ''Goodman 53 of 460'' Other hats that became necessary and popular include the Derby, the straw Boater, and a flat cap with a short brim. In many cases the class, work activity and income could be determined by what kind of hat was on the head of the wearer. For some men, a hat supporting a particular sport or team was important. The Derby or Bowler hat was designed by William and Thomas Bowler, brother shopkeepers in 1849. ''(Goodman 55 of 460)'' It cost less than a top hat but  lasted longer and was soon worn by middle class bankers and clerks. Straw boaters were worn by the aristocracy for casual events and working class factory workers and agricultural laborers. By 1901 working class men had changed their preference to the flat caps which became the most popular hat for the workers. Headdress for men was an essential part of dress for the entire period of the Victorian age, from the 1830s through the end of the century. Judith Flanders describes the hats worn by men in London, <blockquote>It is difficult to bear in mind the importance of hats as not only markers of class and income, but also as indicators of respectability. [509–519] [George Augustus] Sala commented that "every" man throughout the history of the world "must, necessarily and habitually, wear some kind of covering to his head". Postmen wore hats, small children wore hats, field labourers and market gardeners wore hats, cricketers, skaters — all sportsmen — wore hats. It was, self-evidently, impossible to go outdoors without one. ... Those in professional occupations wore pot hats, as did clerks and all those with pretensions to middle-class status. Even doctors' delivery boys wore battered hand-me-down pot hats: "the nap rusty, the band a mournful strip of tarnished lace; but still a Hat", which "stamps him as being associated, in however slender a manner, with a learned profession". Cloth caps were for labourers, for costers and for boys. ... Artisans wore caps made out of paper, which they folded [510–511] themselves and so could easily replace as they became dirty.<ref name=":23" /> (509–511 [of 972]) </blockquote>A pot-hat is a kind of derby or bowler in men's hats. (Lewandowski, 237) ==== Original Text ==== Hats were crucial to a respectable appearance for both men and women. The top hat, for example, was standard formal wear for upper- and middle-class men.<ref name=":4">{{Cite book |last=Steele |first=Valerie |url=https://archive.org/details/fashioneroticism0000stee |title=Victorian Fashion. Fashion and Eroticism: Ideals of Feminine Beauty from the Victorian Era to the Jazz Age |publisher=Oxford University Press |year=1985 |isbn=978-0-19-503530-8 |pages=[https://archive.org/details/fashioneroticism0000stee/page/51 51]–84 |url-access=registration}}</ref> For women, the styles of hats changed over time and were designed to match their outfits. === Women's Hats === For a discussion of the history of plumes and feathers, see [[Social Victorians/Victorian Things#Ostrich Plumes and Prince of Wales's Feathers|Ostrich Plumes and Prince of Wales's Feathers in ''Victorian Things'']]. ==== Original Wikipedia Text ==== During the early Victorian decades, hats were modest in size and design, straw and fabric bonnets being the popular choice. [[Poke bonnet]]s, which had been worn during the late [[Regency period]], had high, small crowns and brims that grew larger until the 1830s, when the face of a woman wearing a poke bonnet could only be seen directly from the front. They had rounded brims, echoing the rounded form of the bell-shaped hoop skirts. Bonnets shrunk at the end of the 1860s and moved to a perched position in the early 1870s as hairstyles grew in scale and intricacy. This led to the popularization of hats, which became the headwear of choice for the remainder of the Victorian era.<ref name="g4223">{{cite book |last=Cunnington |first=Cecil Willett |title=English Women's Clothing in the Nineteenth Century |date=1990-05-01 |publisher=Courier Corporation |isbn=0-486-26323-1 |publication-place=New York |page=}}</ref> [[File:The_London_and_Paris_ladies'_magazine_(Apr_1885)_03.png|thumb|Flower pot style hat of 1885.]] The 1880s saw a hat inspired by the top hat for women known as the flowerpot hat, and the 1890s saw the popularity of the boater. The hats of the late Victorian era were covered with elaborate creations of silk flowers, ribbons, and above all, exotic plumes; hats sometimes included entire exotic birds that had been stuffed. Many of these plumes came from birds in the Florida everglades, which were nearly made entirely extinct by overhunting. By 1899, early environmentalists like [[Adeline Knapp]] were engaged in efforts to curtail the hunting for plumes. By 1900, more than five million birds a year were being slaughtered, and nearly 95 per cent of Florida's shore birds had been killed by [[Plume hunting|plume hunter]]s.<ref>{{cite web|title=Everglades National Park|url=https://www.pbs.org/nationalparks/parks/everglades/|archive-url=https://web.archive.org/web/20090927085907/http://www.pbs.org/nationalparks/parks/everglades/|url-status=dead|archive-date=27 September 2009|publisher=PBS|access-date=7 November 2011}}</ref> == Shoes == The women's shoes of the early Victorian period were narrow and heelless, in black or white satin. By 1850s and 1860s, they were slightly broader with a low heel and made of leather or cloth. Ankle-length laced or buttoned boots were also popular. From the 1870s to the twentieth century, heels grew higher and toes more pointed. Low-cut pumps were worn for the evening.<ref name=":4" /> == Cosmetics == [[Victorian-era cosmetics]] were typically less obvious than ours. However, small amounts of pale face powder or powdered blush were widely used.<ref>{{Cite book |last=Goodman |first=Ruth |title=How to be a Victorian |date=2014 |publisher=Penguin Books |isbn=978-0-670-92136-2 |location=London}}</ref> Tints were sometimes added to face creams. Some cosmetics contained toxic or caustic ingredients like lead, mercury, ammonia, and arsenic {{Citation needed|date=October 2025}}. Hair color ==Mourning black== {{See also |Mourning stationery}} [[File:The royal children in mourning Mar 1862.jpg|thumb|Victoria's five daughters (Alice, Helena, Beatrice, Victoria and Louise), photographed wearing mourning black beneath a bust of their late father, Prince Albert (1862)]] [[File:Mourning dress MET 50.40.3a-b front CP4.jpg|alt=Black Victorian mourning dress|thumb|Mourning Dress, 1894–95]] In Britain, black is the colour traditionally associated with mourning for the dead. The customs and etiquette expected of men, and especially women, were rigid but evolving during much of the Victorian era. The expectations depended on a complex hierarchy of close or distant relationship with the deceased. (Davidoff) The closer the relationship, the longer the mourning period and the wearing of black. The wearing of full black was known as First Mourning, which had its own expected attire, including fabrics, and an expected duration of 4 to 18 months. Following the initial period of First Mourning, the mourner would progress to Second Mourning, a transition period of wearing less black, which was followed by Ordinary Mourning, and then Half-mourning. Some of these stages of mourning were shortened or skipped completely if the mourner's relationship to the deceased was more distant. Half-mourning was a transition period when black was replaced by acceptable colours such as lavender and mauve, possibly considered acceptable transition colours because of the tradition of [[Church of England]] (and [[Catholic Church|Catholic]]) clergy wearing lavender or mauve [[Stole (vestment)|stoles]] for funeral services, to represent the [[Passion (Christianity)|Passion of Christ]].<ref>{{cite web|title=The Colors of the Church Year|url=http://fullhomelydivinity.org/articles/colors.htm|publisher=Consortium of Country Churches|access-date=6 November 2011|archive-date=13 November 2011|archive-url=https://web.archive.org/web/20111113075214/http://fullhomelydivinity.org/articles/colors.htm|url-status=dead}}</ref> The mourning dress worn by Queen Victoria (below, right) "shows the traditional touches of mourning attire, which she wore from the death of her husband, Prince Albert (1819–1861), until her own death."<ref>{{Cite web|url=https://www.metmuseum.org/art/collection/search/155839?&searchField=All&sortBy=Relevance&deptids=8&ft=queen+victoria&offset=0&rpp=20&amp;pos=2|title=Mourning Dress, 1894–95|last=The Metropolitan Museum of Art|date=7 September 2019|website=The Metropolitan Museum of Art|access-date=7 September 2019}}</ref> Dating from 1894–95, Queen Victoria wore this dress as a result of the death of the eldest son of the Prince and Princess of Wales, Eddy, in line to the throne. === Norms for mourning=== ''Manners and Rules of Good Society, or, Solecisms to be Avoided'' (London, Frederick Warne & Co., 1887) gives clear instructions, such as the following:<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–83}}</ref> {| class="wikitable" |- ! Relationship to deceased !! First mourning !! Second mourning !! Ordinary mourning !! Half-mourning |- | Wife for husband || 1-year, 1-month; [[bombazine]] fabric covered with [[Crape|crepe]]; [[widow's cap]], [[lawn cuff]]s, collars || 6 months: less crepe || 6 months: no crepe, silk or wool replaces bombazine; in last 3 months jet jewellery and ribbons can be added || 6 months: colours permitted are grey, lavender, mauve, and black-and-grey |- | Daughter for parent || 6 months: black with black or white crepe (for young girls); no linen cuffs and collars; no jewellery for first 2 months || 4 months: less crepe || – || 2 months as above |- | Wife for husband's parents || 18 months in black bombazine with crepe || – || 3 months in black || 3 months as above |- | Parent for son- or daughter-in-law's parent || – Black armband in representation of someone lost || – || 1-month black || – |- | Second wife for parent of a first wife || – || – || 3 months black || – |} The complexity of these etiquette rules extends to specific mourning periods and attire for siblings, step-parents, aunts and uncles distinguished by blood and by marriage, nieces, nephews, first and second cousins, children, infants, and "connections" (who were entitled to ordinary mourning for a period of "1–3 weeks, depending on level of intimacy"). Men were expected to wear mourning black to a lesser extent than women, and for a shorter mourning period. After the mid-19th century, men would wear a black hatband and black suit, but for only half the prescribed period of mourning expected of women. Widowers were expected to mourn for a mere three months, whereas the proper mourning period expected for widows was up to four years.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|pages=378–9}}</ref> Women who mourned in black for longer periods were accorded great respect in public for their devotion to the departed, the most prominent example being Queen Victoria herself. It was not uncommon for a widow who did not remarry to wear half-mourning for the rest of her life, except when another death necessitated full mourning. For example, Alexandra, Princess of Wales wore half-mourning for the rest of her life after her eldest son Eddy died in 1894. Empress Elisabeth of Austria did the same, as did Empress Eugénie of France. They reverted to full mourning when appropriate, but they never wore less than half-mourning after their sons' deaths. Women with lesser financial means tried to keep up with the example being set by the middle and upper classes by dyeing their daily dress. Dyers made most of their income during the Victorian period by dyeing clothes black for mourning.<ref>{{cite book|last=Flanders|first=Judith|title=The Victorian House|year=2003|publisher=Harper Perennial|location=London|isbn=0-00-713189-5|page=341}}</ref> == Technological advancement == The technological changes that affected the manufacture and consumption of clothing in the Victorian age included the following: * the mass production of fabrics — for example, "by the early 1850s there were thousands of steam-powered looms churning out millions of miles of fabric every year"  [62] * the invention of aniline dyes, which were much more vibrantly colored and resistant to fading than the natural dyes that had been used. — . Invented by chemist [[William Henry Perkin]] in 1856, the first aniline dye mauveine (or mauve) "wash[ed] the fashionable landscape in a haze of purple."<ref name=":22">{{Cite book|title=The Dress Diary: Secrets from a Victorian Woman's Wardrobe|last=Strasdin|first=Kate|publisher=Pegasus Books|year=2023|location=New York, New York}}</ref> (247) Other intense and, to the Victorians, intensely exciting colors followed, but the new synthetic additions to fabric sometimes included chemicals harmful to their wearers. For example, a "bright-magenta hue was achieved by adding arsenical-based chemicals to existing aniline dyes, brightening the already luminous shades – but these left residues themselves, along with a toxic labour trail in their wake."<ref name=":22" /> (255) Perhaps the most famous of these is arsenic green, used on fabrics, wallpapers, and trim: "The craze for artificial foliage to adorn the heads and dresses of women of fashion in the mid-nineteenth century had seen the proliferation of flower workshops, where young women in their hundreds laboured to produce the lifelike green leaves and blooms that would make a fetching headdress or would trail becomingly across the bodice of a gown. The lushness of the green was achieved by the application of a powder, a pigment that was created by mixing copper and the highly toxic chemical, arsenic trioxide. The physical effects of working with this poisonous compound were horrific. Contemporary medical drawings depict the green hue of the skin and dreadful open lesions on the hands of the maker, whilst the daily gradual ingestion of the powder by the flower girls was eventually fatal."<ref name=":22" /> (255) * the invention of a sewing machine that could be used in the home. Although sewing machines were already in use in the clothing industry, in 1858 Isaac Merritt Singer began to sell "lightweight domestic machines" for home sewing, radically increasing women's control over their own dress.<ref name=":24">{{Cite book|title=Victorian Fashions for Women|last=Kay|first=Fiona|last2=Storey|first2=Neil R.|publisher=Pen & Sword History|year=2022|isbn=978 1 39900 416 9|location=Yorkshire and Philadelphia|pages=}}</ref> (91 [of 298]) * the spread of journalism for women and fashion journalism * the Jacquard loom: "French inventor Joseph-Marie Jacquard’s loom attachment mechanized the weaving of figured fabrics by 1804."<ref name=":25" /> (454) Perhaps not at the same scale as these but as important in 1850s designs was a technology that turned iron into steel, which could then be drawn into fine wires.<ref name=":3">{{Cite book|title=The Culture of Fashion|last=Breward|first=Christopher|publisher=Manchester University Press|year=1995|pages=145–180}}</ref> Steel was refined to a malleable state so that thin blades could be curved into concentric circles (called hoops) and connected with wires to form the cage. Technological advancements not only influenced the economy but brought a major change in the fashion styles worn by men and women. As the Victorian era was based on the principles of gender, race and class.<ref>{{cite journal|last1=Graham|first1=P|title=The Victorian Era|url=https://archive.org/details/in.ernet.dli.2015.261548|journal=Digital Library of India}}</ref> Much advancement was in favor of the upper class as they were the ones who could afford the latest technology and change their fashion styles accordingly. In 1830s there was introduction of horse hair crinoline that became a symbol of status and wealth as only the upper-class women could wear it. In 1850s there were more fashion technological advancements hence 1850s could rightly be called a revolution in the Victorian fashion industry such as the innovation of artificial cage crinoline that gave women an artificial hourglass silhouette without layers of petticoats, which was lighter and more hygienic.<ref>{{cite book|last1=Shrimpton|first1=J|title=Victorian Fashion|publisher=Bloomsbury Shire Publications}}</ref> Synthetic dyes, such as [[mauveine]] (aniline purple), were introduced in 1856, adding bright colours to garments. In 1855's ''[[Haute couture]]'' was introduced as tailoring became more mainstream in years to follow.<ref>{{cite book|last1=Aspelund|first1=Karl|title=Fashioning Society|publisher=Fairchild Books}}</ref> Charles Frederick Worth, a prominent English designer, became popular amongst the upper class though its city of destiny always is Paris. Haute couture became popular at the same time that sewing machines were invented.<ref name="Haute Couture">{{cite book|last1=Martin|first1=Richard|last2=Koda|first2=Harold|title=Haute Couture|publisher=The Metropolitan Museum of Art}}</ref> Princess [[Eugénie de Montijo|Eugenie]] of France wore the Englishman dressmaker, Charles Frederick Worth's couture and he instantly became famous in France though he had just arrived in Paris a few years ago. In 1855, Queen Victoria and Prince Albert of Britain welcomed [[Napoleon III]] and Eugenie of France to a full state visit to England. Eugenie was considered a fashion icon in France. Queen Victoria, who had been the fashion icon for European high fashion, was inspired by Eugenie's style and the fashions she wore.{{Citation needed|date=October 2025}} Later, Queen Victoria also appointed Charles Frederick Worth as her dress maker and he became a prominent designer amongst the European upper class. Charles Frederick Worth is known as the father of the haute couture as later the concept of labels were also invented in the late 19th century as custom, made to fit tailoring became mainstream.<ref>{{cite book|last1=Saillard|first1=Olivier|last2=Zazzo|first2=Anne|title=Paris Haute Couture|publisher=Skira Flammarion}}</ref> By the 1860s, when made-to-fit tailoring was popular in Europe, crinolines were considered impractical. In the 1870s, women preferred more slimmer silhouettes, hence bodices grew longer and the polonaise, a skirt and bodice made together, was introduced. In 1870s the Cuirass Bodice, a piece of armour that covers the torso and functions like a corset, was invented. Towards the end of Victoria's reign, dresses were flared naturally as crinolines were rejected by middle-class women. Designers such as Charles Frederick Worth were also against them. All these inventions and changes in fashion led to women's liberation as tailored looks improved posture and were more practical.<ref name="Haute Couture"/> dressmakers, couturiers, modistes == Home decor == {{main|Victorian decorative arts}} Home decor started spare, veered into the elaborately draped and decorated style we today regard as Victorian, then embraced the retro-chic of [[William Morris]] as well as pseudo-[[Japonaiserie]]. == Myths and Oversimplifications == === Modesty === {{main|Victorian morality}} {{Original research|section|date=May 2008}} [[File:1868-skirt-lengths-girl-ages-Harpers-Bazar.gif|thumb|upright|"The proper length for little girls' skirts at various ages", from ''[[Harper's Bazaar]]'', showing a 1900 idea of how the hemline should descend towards the ankle as a girl got older]]Many myths and exaggerations about the period persist to the modern day. Examples include the idea of men's clothing is seen as formal and stiff, women's as elaborate and over-done; clothing covered the entire body, and even the glimpse of an ankle was scandalous. Critics contend that [[corset]]s constricted women's bodies and lives. Homes are described as gloomy, dark, cluttered with massive and over-ornate furniture and proliferating [[bric-a-brac]]. Myth has it that even piano legs were scandalous, and covered with tiny [[pantalette]]s. === Tight Lacing === Tight-lacing, which was not possible until the development of the grommet in 1828, was famously controversial in the Victorian age, generating many column inches of profitable newspaper copy, in part because it was (and still is) fetishistic and subversive in that adolescent girls used it as a means of rebellion and upper-working- or lower-middle-class shop girls saw it as a means of upward mobility.<ref name=":21">{{Cite book|title=Fashion and Fetishism: Corsets, Tight-Lacing and Other Forms of Body-sculpture|last=Kunzle|first=David|publisher=History Press|year=2013|isbn=978 0 7524 9545 3|location=Stroud, Gloucestershire|pages=}}</ref> (71 [of 1182]) No evidence exists that tight lacing was widespread or particularly dangerous.<ref name=":21" /> () In truth, men's formal clothing may have been less colourful than it was in the previous century, but brilliant [[waistcoat]]s and [[cummerbund]]s provided a touch of colour, and [[smoking jacket]]s and [[robe|dressing gown]]s were often of rich Oriental [[brocade]]s. This phenomenon was the result of the growing textile manufacturing sector, developing mass production processes, and increasing attempts to market fashion to men.<ref name="shannon597"/> Corsets stressed a woman's sexuality, exaggerating hips and bust by contrast with a tiny waist. Women's [[evening gown]]s bared the shoulders and the tops of the breasts. The [[jersey dress]]es of the 1880s may have covered the body, but the stretchy novel fabric fit the body like a glove.<ref>{{cite book |last=Gernsheim |first=Alison |title=Victorian & Edwardian Fashion: A Photographic Survey |year=1981 |publisher=Dover Publications |location=New York |page=65|edition=New |isbn=0-486-24205-6}}</ref> Home furnishing was not necessarily ornate or overstuffed. However, those who could afford lavish draperies and expensive ornaments, and wanted to display their wealth, would often do so. Since the Victorian era was one of increased social mobility, there were ever more ''[[nouveaux riches]]'' making a rich show. The items used in decoration may also have been darker and heavier than those used today, simply as a matter of practicality. London was noisy and its air was full of [[soot]] from countless coal fires. Hence those who could afford it draped their windows in heavy, sound-muffling curtains, and chose colours that didn't show soot quickly. When all washing was done by hand, curtains were not washed as frequently as they might be today. There is no actual evidence that piano legs were considered scandalous. Pianos and tables were often draped with [[shawl]]s or cloths—but if the shawls hid anything, it was the cheapness of the furniture. There are references to lower-middle-class families covering up their [[pine]] tables rather than show that they couldn't afford [[mahogany]]. The piano leg story seems to have originated in the 1839 book, ''A Diary in America'' written by Captain [[Frederick Marryat]], as a satirical comment on American prissiness.<ref>{{cite book |last1=Marryat |first1=C.B. |title=A Diary in America: With Remarks on Its Institutions |date=1839 |publisher=Longman, Orme, Brown, Green, and Longmans |location=London, England |volume=2 |pages=246–247 |url=https://books.google.com/books?id=2-VEAAAAIAAJ&pg=PA246}} From pp. 246-247: "I was requested by a lady to escort her to a seminary for young ladies, and on being ushered into the reception-room, conceive my astonishment at beholding a square piano-forte with four ''limbs''. However, that the ladies who visited their daughters, might feel in its full force the extreme delicacy of the mistress of the establishment, and her care to preserve in their utmost purity the ideas of the young ladies under her charge, she had dressed all these four limbs in modest little trousers, with frills at the bottom of them!"</ref> Victorian manners may have been as strict as imagined—on the surface. One simply did not speak publicly about sex, childbirth, and such matters, at least in the respectable middle and upper classes. However, as is well known, discretion covered a multitude of sins. Prostitution flourished. Upper-class men and women indulged in [[adultery|adulterous]] liaisons. == Gallery == {{gallery |2=A mid-Victorian interior: ''Hide and Seek'' by [[James Tissot]], c. 1877 Image:Winterhalter Elisabeth.jpg|3=Dress designed by [[Charles Frederick Worth]] for [[Elisabeth of Bavaria|Elisabeth of Austria]] painted by [[Franz Xaver Winterhalter]].|4=File:Frith A Private View detail.jpg|5=[[William Powell Frith]]'s painting of 1883 contrasts women's [[Aesthetic dress]] (left and right) with fashionable attire (center).|6=File:Tissot lilacs 1875.jpg|7=Day dress, c. 1875 [[James Tissot]] painting.|8=File:James Abbot McNeill Whistler 011.jpg|9=[[James McNeill Whistler|Whistler]]'s [[Portrait of Lady Meux]], 1882 Image:Jeanna_Samary-Renoir.png|10=[[Pierre-Auguste Renoir|Renoir]]'s portrait of [[Jeanne Samary]] in an [[evening gown]], 1878|11=File:Melville_-_Queen_Victoria.jpg|12=Portrait by [[Alexander Melville (artist)|Alexander Melville]] of [[Victoria of the United Kingdom|Queen Victoria]], 1845|13=File:Henry Treffry Dunn Rossetti and Dunton at 16 Cheyne Walk.jpg|14=An artistic interior: [[Dante Gabriel Rossetti]] reading to [[Theodore Watts-Dunton]] in the drawing room at No. 16 [[Cheyne Walk]], 1882|15=File:Punch - Masculine beauty retouched1.png|16=Men's swimwear: Cartoon from ''[[Punch (magazine)|Punch]]'' by [[George du Maurier]]}} == See also == * [[Emily Clapham]] * [[Victorian decorative arts]] * [[Victorian dress reform]] * [[Victorian morality]] * [[Victoriana]] * [[Women in the Victorian Era]] * [[Charles Frederick Worth]] === Time periods === * [[1830s in fashion]] * [[1840s in fashion]] * [[1850s in fashion]] * [[1860s in fashion]] * [[1870s in fashion]] * [[1880s in fashion]] * [[1890s in fashion]] === Women's clothing === * [[Corset]] * [[Corset controversy]] * [[Tightlacing]] * [[Bloomers (clothing)|Bloomers]] * [[Bodice]] === Contemporary interpretations === * [[Steampunk]] * [[Neo-Victorian]] * [[Lolita Fashion|Lolita]] == References == {{Reflist}} == Further reading == *{{cite book |author=Phipps, Elena| title= ''From Queen to Empress: Victorian dress 1837-1877'' | location=New York | publisher=The Metropolitan Museum of Art | year=1988 | isbn=0870995340| url= http://libmma.contentdm.oclc.org/cdm/compoundobject/collection/p15324coll10/id/69547/rec/235 | display-authors=etal}} * Sweet, Matthew – ''Inventing the Victorians'', St. Martin's Press, 2001 {{ISBN|0-312-28326-1}} == External links == * [http://www.victorians.co.uk/victorian-fashion Victorian Fashion] {{Webarchive|url=https://web.archive.org/web/20180407223711/http://www.victorians.co.uk/victorian-fashion |date=7 April 2018 }} * [https://www.victorianvoices.net/topics/fashion/index.shtml VictorianVoices.net] – Fashion articles and illustrations from Victorian periodicals; extensive fashion image gallery * [http://www.cracked.com/article_19575_5-ridiculous-sex-myths-from-history-you-probably-believe.html Victorian myths] * [http://www.victorianstation.com/lifestylemenu.htm Victorian fashion, etiquette, and sports] {{Webarchive|url=https://web.archive.org/web/20180103162620/http://www.victorianstation.com/lifestylemenu.htm |date=3 January 2018 }} * [http://www.thesmartset.com/article/article12180701.aspx Background on "A Diary in America"] * [http://www.mccord-museum.qc.ca/en/keys/webtours/VQ_P2_17_EN.html Form and Fashion] — the evolution of women's dress during the 19th century (many photographs) * [http://www.mccord-museum.qc.ca/en/keys/games/jeu2/ Educational Game: Mix and Match] — build a 19th-century dress using a virtual mannequin * {{cite web |publisher= [[Victoria and Albert Museum]] |url= http://www.vam.ac.uk/content/articles/v/victorian-dress-at-v-and-a/ |title= Victorian Dress |work= Fashion, Jewellery & Accessories |date= 14 January 2011 |access-date= 2011-04-03}} *[http://cv.vic.gov.au/stories/creative-life/fashion-detective-fashion-fiction-and-forensics/ Fashion detective: Fashion, Fiction and Forensics in nineteenth century Australian fashion] on Culture Victoria {{Timeline of clothing and fashion|state=collapsed}}{{Victorian era|state=collapsed}} [[Category:Victorian fashion| ]] [[Category:19th-century fashion|*]] [[Category:1900s fashion]] [[Category:History of Western fashion]] [[Category:19th century in the arts]] =From ''Women in the Victorian era''= ===Victorian women's fashion=== {{Multiple issues|{{tone|date=March 2023}} {{more footnotes needed|date=March 2023}}|section=y}}{{Further|Victorian fashion}} The ideal Victorian woman was pure, chaste, refined, and modest. This ideal was supported by etiquette and manners. The etiquette extended to the pretension of never acknowledging the use of undergarments (sometimes generically referred to as "unmentionables"). The discussion of such a topic, it was feared, would gravitate towards unhealthy attention on anatomical details. As one Victorian lady expressed it: "[those] are not things, my dear, that we speak of; indeed, we try not even to think of them", in contrast to current norms.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=20}}</ref> The pretence of avoiding acknowledgement of anatomical realities met with embarrassing failure on occasion. In 1859, the Hon. Eleanor Stanley wrote about an incident where the [[Louisa Cavendish, Duchess of Devonshire|Duchess of Manchester]] moved too quickly while manoeuvring over a [[stile]], tripping over her large [[hoop skirt]]: {{blockquote|[the Duchess] caught a hoop of her cage in it and went regularly head over heels lighting on her feet with her cage and whole petticoats above, above her head. They say there was never such a thing seen – and the other ladies hardly knew whether to be thankful or not that a part of her undergarments consisted in a pair of scarlet tartan [[knickerbockers (clothing)|knickerbockers]] (the things Charlie shoots in) which were revealed to the view of all the world in general and the [[Aimable Pélissier|Duc de Malakoff]] in particular".<ref>{{cite book|last=Cunnington|first=C. Willett|title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations|year=1990|publisher=Dover Publications|isbn=978-0-486-26323-6|pages=20–1}}</ref>}} However, despite the fact that Victorians considered the mention of women's undergarments in mixed company unacceptable, men's entertainment made great comedic material out of the topic of ladies' [[bloomers (clothing)|bloomers]], including men's magazines and music hall skits.<ref>{{cite book |last=Cunnington |first=C. Willett |title=English Women's Clothing in the Nineteenth Century: A Comprehensive Guide with 1,117 Illustrations |publisher=Dover Publications |year=1990 |isbn=978-0-486-26323-6 |pages=22}}</ref> Victorian women's clothing followed trends that emphasised elaborate dresses, skirts with wide volume created by the use of layered material such as [[crinoline]]s, hoop skirt frames, and heavy fabrics. Because of the impracticality and health impact of the era's fashions, a [[Victorian dress reform|dress reform movement]] began among women. The ideal silhouette of the time demanded a narrow waist, which was accomplished by constricting the abdomen with a laced [[corset]]. While the silhouette was striking, and the dresses themselves were often exquisitely detailed creations, the fashions were cumbersome. At best, they restricted women's movements and at worst, they had a harmful effect on women's health. Physicians turned their attention to the use of corsets and determined that they caused several medical problems: compression of the thorax, restricted breathing, organ displacement, poor circulation, and prolapsed uterus.<ref name="O'Connor"/> Articles advocating the reform of women's clothing by the British National Health Society, the Ladies' Dress Association, and the [[Rational Dress Society]] were reprinted in ''The Canada Lancet'', Canada's medical journal. In 1884, Dr J. Algernon Temple of Toronto even voiced concern that the fashions were having a negative impact on the health of young women from the working classes. He pointed out that a young working-class woman was likely to spend a large part of her earnings on fine hats and shawls, while "her feet are improperly protected, and she wears no flannel petticoat or woollen stockings".<ref name="O'Connor"/> [[File:Bloomers.jpg|thumb|1850s illustration of a woman wearing [[bloomers]]]] [[Florence Pomeroy]], Lady Haberton, was president of the Rational Dress movement in Britain. At a National Health Society exhibition held in 1882, Viscountess Haliburton presented her invention of a "[[divided skirt]]", which was a long skirt that cleared the ground, with separate halves at the bottom made with material attached to the bottom of the skirt. She hoped that her invention would become popular by supporting women's freedom of physical movement, but the British public was not impressed by the invention, perhaps because of the negative "unwomanly" association of the style with the American [[Bloomers]] movement.<ref>{{cite book|last=Murray|first=Janet Horowitz|title=Strong-Minded Women and Other Lost Voices from 19th Century England|year=1982|publisher=Pantheon Books|location=New York|isbn=0-394-71044-4|pages=[https://archive.org/details/strongmindedwome00jane/page/68 68–70]|url=https://archive.org/details/strongmindedwome00jane/page/68}}</ref> [[Amelia Jenks Bloomer]] had encouraged the wearing of visible bloomers by feminists to assert their right to wear comfortable and practical clothing, but it was no more than a passing fashion itself among radical feminists. The movement to reform women's dress would persist and have long-term success, however; by the 1920s, [[Coco Chanel]] was successful at selling a progressive, far less restrictive silhouette that abandoned the corset and raised hemlines. The new silhouette symbolised modernism for trendy young women and became the 20th century standard. Other Paris designers continued reintroducing pants for women and the trend was gradually adopted over the next century. Fashion trends, in one sense, travelled "full circle" over the course of the Victorian era. The popular women's styles during the [[Georgian era]], and at the very beginning of Victoria's reign, emphasized a simple style influenced by flowing gowns worn by women in [[Ancient Greek clothing|Ancient Greece]] and [[Clothing in ancient Rome|Rome]]. The [[Empire waist]] silhouette was replaced by a trend towards ornate styles and an artificial silhouette, with the restrictiveness of women's clothing reaching its low point during the mid-century passion for narrow corseted waists and hoop skirts. The iconic wide-brimmed women's hats of the later Victorian era also followed the trend towards ostentatious display. Hats began the Victorian era as simple [[Bonnet (headgear)|bonnets]]. By the 1880s, milliners were tested by the competition among women to top their outfits with the most creative (and extravagant) hats, designed with expensive materials such as silk flowers and exotic plumes such as ostrich and peacock. As the Victorian era drew to a close, however, fashions were showing indications of a popular backlash against excessive styles. Model, actress and socialite [[Lillie Langtry]] took London by storm in the 1870s, attracting notice for wearing simple black dresses to social events. Combined with her natural beauty, the style appeared dramatic. Fashions followed her example (as well as Queen Victoria's wearing of mourning black later in her reign). According to [[Harold Koda]], the former Curator-in-chief of the [[Costume Institute at The Met|Metropolitan Museum of Art's Costume Institute]],<ref>{{cite web|url=http://www.metmuseum.org/about-the-museum/press-room/exhibitions/2014/death-becomes-her|title=Death Becomes Her: A Century of Mourning Attire : October 21, 2014-February 1, 2015|website=Metmuseuim.org|access-date=7 November 2021}}</ref> "The predominantly black palette of [[mourning]] dramatizes the evolution of period silhouettes and the increasing absorption of fashion ideals into this most codified of etiquettes," said Koda, "The veiled widow could elicit sympathy as well as predatory male advances. As a woman of sexual experience without marital constraints, she was often imagined as a potential threat to the social order." ====Evolution of Victorian women's fashion==== <gallery> File:Fashion plate December 1844.jpg|Ladies' December Fashions (1844). Hand-coloured steel engraving from a women's magazine. File:Thegalleryofhmscalcutta james tissot 1876.jpg|''[[The Gallery of HMS Calcutta]]'' by [[James Tissot]] (1876). [[Bustle]]s were fashionable in the 1870s and 1880s. File:Mrs lillie langtry george frederic watts 1880.jpg|''Mrs. Lillie Langtry'' by [[George Frederic Watts]] (1880). File:Five-women-on-queenslander-steps-r.jpg|Fashionable women in [[Queensland]], Australia around 1900. </gallery> {{Short description|Irish writer (born 1963)}} {{Use Irish English|date=August 2025}} {{Use dmy dates|date=August 2025}} {{Infobox writer | name = Darach Ó Scolaí | image = Darach Ó Scolaí.JPG | alt = Man holding prize-winning book | caption = Ó Scolaí in 2019 | birth_name = Darach Ó Scolaí | birth_date = {{Birth date and age|1963|df=y}} | birth_place = [[County Galway]], The Republic of Ireland | death_date = | death_place = | occupation = Writer, artist, publisher | alma_mater = [[University of Galway]] | years_active = 1998–present | genre = Novel, retelling, translation, play, screenplay, illustrated book for children and adults | other_names = | spouse = | children = 3 | awards = [[Awards and Honors received by Darach Ó Scolaí|Awards and Honors]] | signature = | website = }}[[File:Darach Ó Scolaí.JPG|thumb|Darach Ó Scolaí, holding ''Oileán an Órchiste'' (his translation of Robert Louis Stevenson's ''Treasure Island'')]] == Darach Ó Scolaí == Darach Ó Scolaí (<small>Irish:</small> [/ˈda.rax/ /oː/ /sˠkˠoː/l̪ˠəi/]; born 1963<ref>{{Cite web|url=https://portraidi.ie/en/darach-o-scolai/|title=Darach Ó Scolaí|date=20 October 2017|website=Portráidí (Portraits of Irish-Language Writers)|access-date=1 August 2025}}</ref>) is an Irish author who works in a number of genres, from novels, plays and screenplays to illustrated books for children and adults. He began his literary career in 1998 writing screenplays, stage plays, retellings and translations; he began to publish novels in 2008. Ó Scolaí is widely recognized as a leading figure in contemporary Irish literature, known as “one of the most important Irish language writers of his generation”<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=2022|title=? Suil an Daill: Constant Tensions and Shifting Allegiances|url=https://booksirelandmagazine.com/suil-an-daill-constant-tensions-and-shifting-allegiances/|journal=Books Ireland}}</ref> and "one of the great Irish language novelists [duine d’úrscéalaithe móra na Gaeilge]."<ref name=":17" /> His writing has been called “the high literature of the Irish language.”<ref>Ó Coimín, Maitiú. ''Nós'' 2 February 2018). Qtd. in "Táin Bó Cuailnge." ''Leabhar Breac''. Retrieved 25 August 2025.</ref> Much of his fiction is based on a knowledge of traditional Irish tales and narrative practices as well as Irish history. He specializes in literary and [[wikipedia:Historical_fiction|historical fiction]], or as novelist Alan Titley says, Ó Scolaí’s “peak (for now), or at least his greatest imaginative interest, is the historical novel [tá an chuma air gurb é a bhuaic (go fóill), nó ar a laghad, a mhórspéis samhlaíochta, an t-úrscéal staire].”<ref name=":11">{{Cite journal|last=Titley|first=Alan|date=Fall 2020|title=An Stíl Go Deo!: Soather Dharach Uí Scolaí (The style would be forever!: Worker Darach Ó Scolaí)|url=https://www.jstor.org/stable/27046090|journal=Comhar|volume=80, No. 10|pages=27|via=JSTOR}}</ref> His retellings of old stories and tales from their original Middle and Early-Modern Irish into Modern Irish ([[wikipedia:Irish_language|Gaeilge]]) are respected for their accessibility to students and language learners as well as for their artistry. Ó Scolaí also regularly reviews books and lectures and writes on literature and culture. Beyond his writing, Ó Scolaí is a publisher and has co-produced a number of film, television shows and stage plays. == Life == Ó Scolaí was born in Dublin and raised in the Galway [[wikipedia:Gaeltacht#Galway Gaeltacht|Gaeltacht]] (Irish-speaking) regions of Cois Fharraige on the north shore of Galway Bay, in the Republic of Ireland, where he lives now with his wife and children in Lochán Beag (Indreabhán).<ref name=":7">{{Cite journal|date=30 October 2024|title=Duais don úrscéal liteartha is fearr buaite ag Darach Ó Scolaí ag Oireachtas na Samhna|url=https://tuairisc.ie/duais-don-ursceal-liteartha-is-fearr-buaite-ag-darach-o-scolai-ag-oireachtas-na-samhna/|journal=Tuairisc}}</ref><ref>{{Cite journal|last=Ní Scolaí|first=Aifric|date=2024|title=Darach Ó Scolaí|url=https://www.taiscecf.ie/ealaiontoiri?category=Scr%C3%ADbhneoir|journal=Taisce Chois Fharraige}}</ref> He graduated the [[wikipedia:University_of_Galway|University of Galway]] (then University College Galway) with a B.A. in 1983.<ref>{{Cite web|url=https://www.linkedin.com/in/darach-ó-scolaí-20026920/|title=Darach Ó Scolaí|last=Ó Scolaí|first=Darach|date=August 2025|website=LinkedIn}}</ref> === Writing and Publishing === Ó Scolaí writes in Irish ([[wikipedia:Irish_language|Gaeilge]]), his native language, and lives in an area defined for the predominant presence of Irish as the vernacular language, the language spoken at home. Irish was the language of his parents' home and is the language of children as well. He is fluent in Irish and English and conversant in French. None of his works has been translated into English. ==== Leabhar Breac ==== In 1995 Darach Ó Scolaí and his brother Caomhán Ó Scolaí — a [[wikipedia:Typography|typographer]] and designer — founded the publishing house Leabhar Breac at Indreabhán (Inverin), County Galway. Their father “Séamas Ó Scolaí was an editor at An Gúm and worked on the Irish-English dictionary team [bhí a n-athair Séamas Ó Scolaí ina eagarthóir sa Ghúm agus d’oibrigh sé ar fhoireann an fhoclóra Gaeilge-Béarla].”<ref name=":0">{{Cite web|url=https://leabharbreac.com/en/about-us/|title=About Us|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> Darach Ó Scolaí has been publisher and literary editor at Leabhar Breac since its founding. Named for [[wikipedia:An_Leabhar_Breac|An Leabhar Breac (The Speckled Book)]], Leabhar Breac publishing house has more than 140 books in print.<ref name=":0" /> Leabhar Breac aims to publish Irish-language books that meet “a high literary and artistic standard.”<ref name=":0" /> Besides the content, Leabhar Breac is known for the typically "superb [thar cionn]" quality of the design and production of the "physical book [leabhar fisiciúil]."<ref name=":8">{{Cite journal|last=Ní Mhuilneoir|first=Gráinne|date=30 July 2024|title=‘Bláthnaid’ – leabhar álainn i sraithín álainn faoi mhná|url=https://tuairisc.ie/blathnaid-leabhar-alainn-i-sraithin-alainn-faoi-mhna/|journal=Tuairisc}}</ref> Its books regularly win awards for literary and artistic quality. Leabhar Breac also publishes translations for children and adults from various early versions of Irish as well as from French and English (and has published translations of books for young readers from Spanish, Catalan, and Italian as well). Leabhar Breac prints its books in Ireland. === Stage and Screen === ==== Rosg ==== In 1998 along with Ciarán Ó Cofaigh,<ref name=":1">{{Cite web|url=http://www.rosg.ie/en/about/History_6/|title=About Us: History|date=July 2025|website=Rosg|access-date=1 August 2025}}</ref> Ó Scolaí co-founded the film and television production company [http://www.rosg.ie/en/ Rosg] and was co-director until 2006. Rosg produced Ó ScolaÍ’s films ''Cosa Nite'' (1999), ''An Leabhar'' (2001) and ''Na Cloigne'' (2010). He left Rosg in 2006 to devote his time to other artistic activities. ==== Ealaín ar Oileán ==== In 2004, along with Val Balance, Ó Scolaí co-founded the annual artists' symposium Ealaín ar Oileán (trans., Art on an Island). The Irish-language symposium was held annually in the Áras Éanna arts and cultural center on Inis Oírr ([[wikipedia:Inisheer|Inisheer]], the smallest of the [[wikipedia:Aran_Islands|Aran Islands]]) from 2004 to 2013. Ó Scolaí was its co-director from its founding<ref>{{Cite web|url=https://ga.wikipedia.org/wiki/Darach_Ó_Scolaí.|title=Darach Ó Scolaí|date=3 February 2024|website=Vicipéid|access-date=1 July 2025}}</ref> until 2013. Besides being its co-director, Ó Scolaí has taken part in this conference as an artist<ref>{{Cite journal|date=16 January 2005|title=Darach Ó Scolaí|url=https://web.archive.org/web/20050116163252/http://bliainiris.com/authors/darach_oscolai.html|journal=Bliainiris}}</ref> and writer<ref name=":2">{{Cite web|url=http://ealainaroilean.ie/ealainaroilean.html|title=The Conference|date=7 September 2013|website=Ealaín ar Oileán|archive-url=https://web.archive.org/web/20130907083744/http://ealainaroilean.ie/ealainaroilean.html|archive-date=7 September 2013|access-date=1 August 2025}}</ref>. ==== Salamandar ==== In 2006 Ó Scolaí founded the stage production company Salamandar and directed his own play ''An Braon Aníos''. His plays ''An tSeanbhróg'' (2009) and ''Craos'' (2008) were also produced by Salamandar.<ref name=":19">{{Cite web|url=https://leabharbreac.com/en/product-category/darach-o-scolai/|title=Darach Ó Scolaí|date=2024|website=Leabhar Breac|access-date=1 July 2025}}</ref> == Works == === Novels === * [[wikipedia:An_Cléireach|''An Cléireach'' (trans., ''The Clerk'')]], Leabhar Breac, 2007. The Oireachtas Prize for Literary Fiction, 2007; The Ó Súilleabháin Award (Book of the Year) in 2008, and "named as ‘the best novel since the turn of the Century’ by Comhar."<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-cleireach/|title=An Cléireach - Leabhar Breac - Irish language novel|website=Leabhar Breac|language=en-US|access-date=2025-10-24}}</ref> * ''Na Comharthaí'' (trans., ''The Signs''), Leabhar Breac, 2014. * ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. The Oireachtas Prize for Literary Fiction, 2019.<ref name=":4">{{Cite web|url=https://leabharbreac.com/en/shop/fiction/suil-an-daill/|title=Súil an Daill|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Bódléar'', Leabhar Breac, 2024. The Oireachtas Prize for Literary Fiction, 2024<ref name=":7" />; The Ó Súilleabháin Award (Book of the Year) in 2025; featured in the 2025 Listen-Up Irish Summer Challenge for students of the Irish language.<ref>{{Cite news|url=https://connachttribune.ie/novel-approach-helps-people-learn-irish-in-a-creative-way/|title=Novel approach helps people learn Irish in a creative way|last=Murphy|first=Judy|date=3 October 2025|work=Connaught Tribune|access-date=24 October 2025}}</ref> === Retellings, Translations and Editions === The retellings and translations are into modern Irish. * ''Feis Tigh Chonáin'' (trans., ''The Feast of Conán's House''), Leabhar Breac, 2000; a retelling of a 15^th-century tale from the [[wikipedia:Fenian_Cycle|Fenian Cycle]].<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/feis-tigh-chonain/|title=Feis Tigh Chonáin|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''An Ceithearnach Caolriabhach'' (trans., ''The Narrow-Striped Kern''), Leabhar Breac, 2002; a retelling from c. 1500, also illustrated by Darach Ó ScolaÍ.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/fiction/an-ceithearnach-caolriabhach/|title=An Ceithearnach Caolriabhach|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), Leabhar Breac, 2017, both a modern edition of an 11th-century epic and an annotated edition.<ref name=":3">{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/tain-bo-cuailnge-2-2/|title=Táin Bó Cuailnge|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> "''Táin Bó Cuailnge'' won the Aodán Mac Poilín Memorial Prize 2017."<ref name=":19" /> * ''Deirdre'', Leabhar Breac, 2023, a “picture book for adults” with artist Anastasia Melnykova.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/deirdre/|title=Deirdre|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Part of the [[wikipedia:Ulster_Cycle|Ulster Cycle]], ''Deirdre'' is a retelling of the story of possibly the most widely known Irish figure from the early tales and sagas.<ref>{{Cite book|title=A Dictionary of Celtic Mythology|last=MacKillop|first=James|publisher=Oxford University Press|year=2004|isbn=9780198609674|pages=181}}</ref> * ''Bláthnaid'', Leabhar Breac, 2024, a “picture book for adults” with artist Anastasia Melnykova; “one of the great stories of the [[wikipedia:Ulster_Cycle|Ulster Cycle]].”<ref name=":4" /> * ''Sadhbh,'' Leabhar Breac, 2025, a picture book for adult readers, illustrated by Alé Mercado; a retelling of the medieval tale ''Ceasacht Inghine Ghuile (''trans., ''The Complaint of Guile's Daughter'').<ref name=":5">{{Cite web|url=https://leabharbreac.com/en/tales-of-wonder/|title=Tales of Wonder|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Eoghan Béal'', Leabhar Breac, 2025, a picture book for adult readers illustrated by Alé Mercado<ref name=":5" />; a retelling of the medieval tale ''[https://ga.wikipedia.org/wiki/Caithr%C3%A9im_Cellaig Cathréim Ceallaigh]'' from ''The Yellow Book of Leacan.''<ref name=":5" /> === For Young Readers === Ó Scolaí has written illustrated books for young readers (8–10 years old) in two series, the Fionn Series and the Scéalta Staire series, and translated a large number of classics and popular books for children of all ages. The number of these written and translated works suggests a commitment to children and their literacy in Irish. The Fionn Series “is a retelling ... of the great legends of the Fianna for the young Irish readers of today.”<ref name=":6">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/doiteoir-na-samhna/|title=Dóiteoir na Samhna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> [[wikipedia:The_Boyhood_Deeds_of_Fionn|Macgnímartha Finn (The Boyhood Deeds of Fionn)]] is a medieval story in the [[wikipedia:Fenian_Cycle|Fenian Cycle]]. * ''An Bradán Feasa'' (trans., ''The Salmon of Knowledge''), Leabhar Breac, 2010, “shortlisted for the Réics Carlo award 2010.”<ref name=":9">{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/8-9/an-bradan-feasa/|title=An Bradán Feasa|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Dóiteoir na Samhna'' (trans., ''The Halloween Burner''), 2010.<ref name=":6" /> * ''Bodach an Chóta Lachna'' (trans., ''The Churl in the Dun Coat''), 2011.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/|title=Bodach an Chóta Lachna|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> The Scéalta Staire (Historical Stories) series<ref name=":9" /> * ''Mánas Ó Dónaill'', 2000. * ''Seán Ó Néill'', Leabhar Breac, 2000. * ''Gráinne Mhaol Ní Mháille'', Leabhar Breac, 2003. * ''Tadhg Dall Ó hUiginn'', Leabhar Breac, 2003. ==== Translations ==== * Robert Louis Stevenson, ''Oileán an Órchiste'' (trans. of ''Treasure Island''), Leabhar Breac, 2014.<ref>{{Cite journal|date=2025-06-19|title=Oireachtas na Gaeilge|url=https://en.wikipedia.org/w/index.php?title=Oireachtas_na_Gaeilge&oldid=1296394643|journal=Wikipedia|language=en}}</ref> * Robert Louis Stevenson, ''An Fuadach'' (trans. of ''Kidnapped''), Leabhar Breac, 2016. * Clement Clarke Moore, ''Cuairt San Nioclás'' (trans. of ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas"), Leabhar Breac, 2022. '''''The Corto Maltese Graphic Novels''''' Written in Italian by Hugo Pratt and translated by Ó Scolaí, both adults and teenagers read this series of Italian adventure graphic novels.<ref>{{Cite journal|date=2025-07-01|title=Corto Maltese|url=https://en.wikipedia.org/w/index.php?title=Corto_Maltese&oldid=1298285365|journal=Wikipedia|language=en}}</ref> Ó Scolaí's '''translation of ''Corto Maltese''''' was listed in 2017 among "The 30 Irish books that Irish people love."<ref>{{Cite journal|last=Ó Murchú|first=Eoin P.|date=09/06/2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil [The 30 Irish books that Irish people love]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * Hugo Pratt, ''Corto: Port na Farraige Goirt'', Leabhar Breac, 2013. * Hugo Pratt, ''Corto: The Golden House in Samarkand'', 2014. * Hugo Pratt, ''Corto: Na Liopard-Fhir ó Rufiji'' (trans. of ''Corto: The Leopard Men of Rufiji''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: In Ainm Dé Uilthrócairigh'' (trans. of ''Corto: In the Name of God All-Merciful''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Tóraíocht Eile'' (trans. of ''Corto: Another Quest''), Leabhar Breac, 2014. * Hugo Pratt, ''Corto: Sa tSibéir'' (trans. of ''Corto: In Siberia''), Leabhar Breac, 2016. '''''Other Translations for Children''''' Ó Scolaí has translated into Irish six books from the ''Le Pavillon Noir'' (trans., ''Jolly Roger'') series by Alain Surget; four books from the ''Catalan First Steps'' series by Enric Lluch Girbés and the ''Caitlín & Cormac'' series by Joan Carles; three books from the ''Louisette le Taupe'' series by Bruno Heitz, and three books from the ''Loup'' series by Orianne Lallemand. === Plays and Screenplays === ==== Stage Plays ==== Ó Scolaí was writer and director of the original productions of two plays in the ''Trí Bhraon'' (trans., ''Three Drops'') trilogy; ''Coinneáil Orainn'' was directed by Darach Mac Con Iomaire and staged by An Taibhdhearc. All three plays have been published in book form by Leabhar Breac. * ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32553|title=Coinneáil Orainn|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> The first play in the ''Trí Bhraon'' (''Three Drops'') trilogy. [[wikipedia:Taibhdhearc_na_Gaillimhe|An Taibhdhearc]], the national Irish-language theatre of Ireland, toured the country in 2005 with ''Coinneáil Orainn''.<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/coinneail-orainn/|title=Coinneáil Orainn|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Walter Macken Prize, 2005; BBC Stewart Parker Award, 2006.<ref>{{Cite web|url=https://irishplayography.com/person/darach-scola|title=Darach Ó Scolaí|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904.|access-date=25 August 2025}}</ref> * ''Branwen'', 2006, by Darach Ó Scolaí and Ifor ap Glyn, in Irish, Welsh and English, co-produced by Project Arts Centre and Llwyfan Gogledd Cymru, toured the Republic of Ireland and Wales.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32418|title=Branwen|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An Braon'' Aníos (trans., ''Rising Damp''), 2006, directed by Ó Scolaí.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32461|title=An Braon Aníos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The second play in the ''Trí Bhraon'' (''Three Drops'') trilogy. “The Salamandar company toured the country in 2006-07 with this play, and Salamandar also produced a radio version of the play for RTÉ Raidió na Gaeltachta in 2009.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/an-braon-anios/|title=An Braon Aníos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''Craos'' (trans., ''Gluttony''), 2008, directed by Ó Scolaí.<ref name=":13">{{Cite web|url=https://irishplayography.com/play?playid=32867|title=Craos|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> The third play in the ''Trí Bhraon'' (''Three Drops'') trilogy, it toured to Cork and Belfast.<ref name=":13" /> A review of the 2008 Salamander performance in the ''Irish Times'' says, “a humorous play which offers plenty to think about, fine acting, and sparklingly witty dialogue.”<ref>{{Cite web|url=https://leabharbreac.com/en/shop/darach-o-scolai/craos-2/|title=Craos|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> * ''A+E'', 2008, by Ríonach Ní Néill and Darach Ó Scolaí, "dance and music drama," co-produced by Ciotóg and Salamandar.<ref>{{Cite web|url=https://irishplayography.com/play?playid=32962|title=A+E|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> * ''An tSeanbhróg'' (trans., ''The Old Shoe''), 2009, produced by Salamander<ref>{{Cite web|url=https://irishplayography.com/play?playid=33042|title=An tSeanbhróg|date=2025|website=PlayographyIreland: A Comprehensive Database of New Irish Plays Produced Professionally Since 1904|access-date=25 August 2025}}</ref> and staged in the Axis Arts Centre, Dublin, and the Letterkenny Arts Centre. * '''In ''Mhuir Fhíondorcha/The Wine-Dark Sea: The Homer Project'', Ó Scolaí's translation of Homer's Cyclops story, performed at the 2019 IMRAM festival'''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/books/imram-a-festival-celebrating-the-irish-language-1.4047610|title=Imram: a festival celebrating the Irish language. Liam Carson reveals the myths and legends appearing in this year’s programme|last=Carson|first=Liam|date=11 October 2019|work=The Irish Times|access-date=15 October 2025}}</ref> ==== Screenplays ==== * ''Cosa Nite'' (trans., ''Washed Feet''), short film, 1998 (dir. Dearbhla Walsh, prod. Ciarán Ó Cofaigh, Rosg); "a prose version of ''Cosa Nite'' was published (Rosg 2000)."<ref name=":9" /> Nominated for an Irish Film and Television Award.<ref>{{Citation|title=Cosa Nite (Short 1998) - Awards - IMDb|url=https://www.imdb.com/title/tt0191917/awards/|accessdate=2025-08-25|language=en-US}}</ref> * ''Na Glúnta'' (trans., ''The Generations''), 2001<ref>{{Cite web|url=https://www.iftn.ie/production/production_companies/production_sub/feature/?act1=record&aid=70&rid=3917&tpl=filmography_dets&only=1&force=1|title=Na Glúnta {{!}} The Irish Film & Television Network|website=www.iftn.ie|access-date=2025-08-25}}</ref>, co-directors Ciarán Ó Cofaigh & Darach Ó Scolaí, prod. Ciarán Ó Cofaigh, Rosg. * ''An Leabhar'' (trans., ''The Book''), short film, 2000, (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg) Rosg, 2000.<ref>{{Citation|title=An Leabhar|url=https://www.imdb.com/title/tt0963767/|publisher=Bord Scannán na hÉireann / The Irish Film Board, ROSG|accessdate=2025-08-25|first=Robert|last=Quinn|others=Colm O&apos;Maonlai, Peadar O&apos;Treasaigh, Diarmuid Mac an Adhastair}}</ref> * ''Na Cloigne'' [trans., The Heads], 3-episide series, 2010 (dir. Robert Quinn, prod. Ciarán Ó Cofaigh, Rosg), TG4.<ref>{{Cite web|url=https://www.imdb.com/title/tt1607924/|title=Na cloigne|date=2010|website=IMDb|access-date=25 August 2025}}</ref> === Nonfiction === Ó Scolaí's essays and lectures are published and his interviews are broadcast regularly, making for a large body of nonfiction critical and analytical work. Here are a few, almost all published in [https://comhar.ie/iris/scribhneoiri/darach-o-scolai/ Comhar]: * “Ceol Ciúin na nÉagmaise” (trans., “The Silent Music of Absence ['''the Fall?''']”), an essay on the 2014 Nobel Prize winner for literature, [[wikipedia:Patrick_Modiano|Patrick Modiano]], ''Comhar'', December 2014. * The Ó Cadhain Lecture: [https://leachtaiuichadhain.clo.ie/leachtai/2014 “Cuimhne agus Díchuimhne (trans., “Memory & Forgetfulness"]), 2014. * “Rithim agus Réim” ("Rhythm and Register"), a public lecture in the University College Dublin lecture series “Ó Thrácht go Twitter” (trans., "From Talk to Twitter"), 2014. * Review of Pádraig Ó Cíobháin’s ''Dréachta Chrích Fodla'', '''Comhar?, ??'''. * “Na Geilt i mBun an Tí” (trans., "The Madmen in Charge"), a talk at the Merriman Winter School, Comhar April 2012.<ref name=":18">{{Cite web|url=http://darachoscolai.ie/beathaisneis.html|title=Darach Ó Scolaí: Beathaisnéis|website=darachoscolai.ie|access-date=2025-09-26}}</ref> * The EFACIS podcast: Síle Ní Choincheannain talks to Darach Ó Scolaí about the historical novel. == Critical Reception == Ó Scolaí’s style has been called “crisp and elegant, and rich in language while being highly readable,”<ref>{{Cite journal|last=Heussaf|first=Anna|date=Summer 2025|title=Bláthnaid—A tale of love, violence and sorcery retold for readers today|url=https://booksirelandmagazine.com/blathnaid-a-tale-of-love-violence-and-sorcery/|journal=Books Ireland}}</ref> with “an unsurpassed richness and precision of language.”<ref name=":12">{{Cite journal|last=Ó Cróinín|first=Breandán|date=Summer 2025|title=unknown|journal=The Limerick Leader}}</ref> “Whimsical, hilarious, and subtly learned” is how Éilis Ní Dhuibhne described his writing.<ref name=":20" /> === Original Works === Ó Scolaí’s first novel, the 2007 ''An Cléireach'' (''The Clerk'') won two prizes and was described as “one of the great historical novels in the Irish language and among the best books written in the language since the beginning of this century.”<ref name=":12" /> Novelist Alan Titley says, “In ''An Cléireach'' Ó Scolaí creates the Ireland of war in the 17th century more fully than any other Irish writer on the subject of war since ''L’Attaque'' Eoghain Ó Thuairisc around 1798 [In ''An Cléireach'' cruthaíonn Ó Scolaí Éire an chogaidh san 17ú haois níos iomláine ná mar a dhein aon scríbhneoir Gaeilge eile ar ábhar cogaidh ó ''L’Attaque'' Eoghain Uí Thuairisc timpeall ar 1798].”<ref name=":11" />{{rp|25, Col. 1a}} Not all the reviews of this first novel were so positive, however; Proinsias O' Drisceoil says for the Irish Times says,<blockquote>This then is a novel in search of a plot, a story that attempts to attain a significance that eludes it.<ref>{{Cite news|url=https://www.irishtimes.com/news/a-disaffected-clerk-in-the-confederates-1.943070|title=A disaffected clerk in the confederates|last=O' Drisceoil|first=Proinsias|date=5 July 2008|work=The Irish Times|access-date=16 October 2025}}</ref></blockquote> In the ''Oxford Handbook of Modern Irish Fiction'' Pádraig Ó Siadhail analyzes rather than reviews ''An Cléireach'': <blockquote>In ''An Cléireach'', Ó Scolaí revisits the trauma of Cromwellian Ireland. The primary narrative device is once again the first-hand account, in this case by Tadhg Ó Dúbháin, a clerk and quartermaster in the Confederate Army in 1650. We sample the hardships, the friendships, the tensions, the rivalries, and the petty jealousies amongst comrades in arms, including remnants of the Gaelic literary class, as the Confederate soldiers, increasingly a rabble more than a cohesive unit, retreat in advance of Cromwell’s forces. ''An Cléireach'' concludes with the narrator and his family in exile in continental Europe. But along the retreat route, and central to the novel, members of the Confederate army camp, rest up, and tell versions of a story about the keeper of the treasured manuscript "Saltair an Easpaig" (The Bishop’s Psalter). Their versions raise issues about memory construction, the limitations of individual perspectives, personal agendas, and how minor changes in the telling of a story can alter our understanding of history, Thus, ''An Cléireach'' complements ''Fontenoy'' in moving beyond more realistic recreation of a historical event or period to interrogate the notion of history as construct.<ref>{{Cite book|title=The Oxford Handbook of Modern Irish Fiction|last=Ó Siadhail|first=Pádraig|publisher=Oxford University Press|year=2020|isbn=9780198754893|editor-last=Harte|editor-first=Liam|pages=598–99|chapter=Contemporary Irish Fiction}}</ref> </blockquote> Of ''Súil an Daill,'' in ''Nós'', Cathal Seoighe says, "The book deserves a significant place among the collection of high-quality books published in recent years that would make you feel sorry for someone who does not speak Irish [Tá áit shuntasach ag dul don leabhar i measc an chnuasaigh leabhair ar ardchaighdeán a foilsíodh le roinnt blianta anuas a d’fhágfadh trua agat don té atá gan Ghaeilge]."<ref>{{Cite journal|last=Seoighe|first=Cathal|date=09/26/2022|title=‘Dar leathmhagairle an diabhail, is leabhar den scoth é seo!’ ['According to the devil’s half-wit, this is a great book!’]|url=https://nos.ie/cultur/leabhair/dar-leathmhagairle-an-diabhail-is-leabhar-den-scoth-e-seo/|journal=Nós}}</ref> ''Bódléar'', Ó Scolaí's most recent book, is a “beautiful novel. There is magic and craftsmanship in it. A small miracle of a book and it is highly recommended.”<ref>{{Cite web|url=https://leabharbreac.com/bodlear-mioruilt-bheag-de-leabhar/|title=Bódléar: Míorúilt bheag de leabhar (Bódléar: A Small Miracle of a Book)|last=Ní Ghairbhí|first=Róisín|date=2024|website=Leabhar Breac|access-date=1 August 2025}}</ref> Éilis Ní Dhuibhne in the ''Irish Times'' says,<blockquote>what a gem! An affectionately gentle satire of the Irish poetic scene during one creatively fluid 19th-century year, the story focuses on a Maigue poet and schoolteacher who goes on a trip to France and returns with camembert, a cafetiere, ‘Fleurs du Mal’, and a mission to convert the local traditionalists to la modernité. Whimsical, hilarious, and subtly learned, it’s absolutely delightful!<ref name=":20">{{Cite journal|last=Ní Dhuibhne|first=Éilis|date=30 June 2025|title=Éilís Ní Dhuibhne on the best Irish language books of 2025 so far: Including a history of the Gaeltacht Civil Rights Movements, a gem of a novel by Darach Ó Scolaí and Joe McHugh’s entertaining account of learning Irish|url=https://www.irishtimes.com/culture/books/review/2025/06/30/eilis-ni-dhuibhne-on-the-best-irish-language-books-of-2025-so-far/|journal=The Irish Times|pages=22}}</ref></blockquote> === Retellings and Translations === ==== ''Táin Bó Cuailnge'' ==== ''Táin Bó Cuailnge'' [''The Cattle Raid of Cooley''] is a modern edition of an 11th-century epic into modern Irish.<ref name=":3" /> Gearóid Denvir reviewed ''Táin Bó Cuailnge'' for ''Comhar'':<blockquote>Darach Ó Scolaí has ​​achieved a feat in this challenging reworking. He has found a high level of the Irish language to tell his story – as he has done before in his groundbreaking novel An Cléireach (2007, Leabhar Breac) and in his other prose works. This book is a decoration of the language, literature and culture of the Irish language, following the path of the old storytellers and writers and presenting material from the tradition to his own generation according to the understandings of his own time. The book will be a classic that will be of great interest to all readers of the Irish language, both ordinary readers, students, scholars and writers, and there should be a copy in every home in the country. [Tá éacht déanta ag Darach Ó Scolaí san athleagan dúshlánach seo. Tá réim ard den teanga Ghaeilge aimsithe aige lena scéal a inseacht – mar a rinne sé cheana ina úrscéal ceannródaíoch An Cléireach (2007, Leabhar Breac) agus i saothair eile phróis dá chuid. Is maisiú ar an teanga agus ar litríocht agus cultúr na Gaeilge an leabhar seo a leanas conair na seanscéalaithe agus na seanscríobhaithe agus ábhar de chuid an traidisiúin á chur i láthair a ghlúine féin aige de réir thuiscintí a linne féin. Clasaic a bheas sa leabhar a gcuirfidh léitheoirí uilig na Gaeilge, idir ghnáthléitheoirí, mhic léinn, scoláirí agus scríbhneoirí spéis thar na bearta ann, agus ba cheart cóip a bheith i chuile theach sa tír.]<ref name=":15">{{Cite journal|last=Denvir|first=Gearóid|date=April 2018|title=Táin Bó Cuailgne|url=https://comhar.ie/iris/78/4/leirmheas/|journal=Comhar|via=JSTOR}}</ref> </blockquote>Cathal Poirtéir says, "The freshness and richness of Ó Scolaí’s version are a joy …. The author delights us with the linguistic and stylistic richness of the ancient epic in a modern-Irish version that reflects the original’s spirit and language."<ref>{{Cite journal|last=Poirtéir|first=Cathal|date=May/June 2018|title=Leabhair Idir Lámha|url=https://www.jstor.org/stable/26564180|journal=Books Ireland|pages=46–47|via=JSTOR}}</ref>{{rp|47}} Novelist and academic Alan Titley calls Ó Scolaí's "a wonderful gutsy telling" of ''Táin Bó Cuailnge''.<ref>{{Cite news|url=https://www.irishtimes.com/culture/2023/03/11/the-tain-retold-maeve-and-ailills-spat-could-be-out-of-a-soap-opera/|title=The Táin retold: ‘Maeve and Ailill’s spat could be out of a soap opera’|last=Titley|first=Alan|date=11 March 2023|work=The Irish Times|access-date=16 October 2025}}</ref> ==== ''Deirdre'' ==== Marie Whelton, in "Léann Teanga" ("Language Studies"), in the 2024 ''An Reiviú'' says,<blockquote>this version [of ''Deirdre''] by Darach Ó Scolaí succeeds in skillfully capturing and portraying the complexity of gender and power issues in the ‘Deirdre’ tradition [éiríonn leis an leagan seo le Darach Ó Scolaí castacht cheisteanna na hinscne agus na cumhachta i dtraidisiún scéal Dheirdre a ghabháil agus a léiriú go sciliúil]. … There is no doubt that this new version greatly contributes to the legacy of the story and that it revives that legacy thoughtfully and artistically [Níl amhras faoi ach go gcuireann an leagan úr seo go mór le hoidhreacht an scéil agus go ndéanann sé an oidhreacht sin a athbheochan go tuisceanach agus go healaíonta.].<ref name=":16">{{Cite web|url=https://www.tara.tcd.ie/tara8/server/api/core/bitstreams/3c20175a-7631-44b2-8b0f-f454edd712b4/content|title=An Artistic Retelling of Deirdre's Tale and the Defeat of Conor Review of Deirdre or the Ship of Mac Uisnigh by Darach Ó Scolaí [Athinsint Ealaíonta ar Oidhe Dheirdre agus ar Ansmacht Chonchúir Léirmheas ar Deirdre nó Loingeas Mhac Uisnigh le Darach Ó Scolaí]|last=Whelton|first=Marie|date=2024|website=The Review [An Reiviú], Language Studies [Léann Teanga]|access-date=25 September 2025}}</ref></blockquote> === Works for Young Readers === Meadhbh Ní Eadhra said of ''Bodach an Chóta Lachna'' that it was "Beautiful Irish, but easy to understand for young readers."<ref>Ní Eadhra, Meadhbh. In ''Gaelscéal'', qtd. in "Bodach an Chóta Lachna" https://leabharbreac.com/en/shop/oige-en/7-8/bodach-an-chota-lachna/.</ref> == Awards and Honors == Ó Scolaí's works are regularly nominated and make the short list for prizes, an honor in itself, but they are generally not listed here unless they are named as the first-place winner in their category. === Oireachtas Prize === The Oireachtas Prize is the literary prize awarded by [[wikipedia:Oireachtas_na_Gaeilge|Oireachtas na Gaeilge]], the annual arts festival dedicated to Irish language, arts and culture. Darach Ó Scolaí has won the Oireachtas Prize for Literary Fiction three times, once for ''An Cléireach'' (''The Clerk'') in 2007, for ''Súil an Daill'' (''The Eye of the Blind'') in 2021 and for ''Bódléar'' in 2024. * 2007, for ''An Cléireach'' (trans., ''The Clerk'') — “(a special prize commemorating the 400th anniversary of the foundation of Coláiste na nGael in Louvain, awarded under the auspices of the Franciscan Province of Ireland). The prize of €10,000 was the largest prize ever awarded to an Irish language novel [(duais speisialta chomórtha 400 bliain bhunú Choláiste na nGael i Lobháin a bronnadh faoi urraíocht Phroibhinse Phroinsiasach na hÉireann). Ba é an duais €10,000 sin an duais ba mhó a bronnadh riamh ar úrscéal Gaeilge].”<ref name=":18" /> * 2021, for ''Súil an Daill'' (''The Eye of the Blind'') * 2024, for ''Bódléar'' === Ó Shúilleabháin Award, Irish language “Book of the Year” === The first prize of this award includes €5,000 to the publisher and €2,500 to the author of the winning work.<ref name=":10">{{Cite journal|date=15 August 2023|title=20 saothar san iomaíocht do ‘Leabhair Ghaeilge na Bliana 2023’|url=https://tuairisc.ie/20-saothar-san-iomaiocht-do-leabhair-ghaeilge-na-bliana-2023/|journal=Tuairisc}}</ref> * ''An Cléireach'' (''The Clerk'').<ref>{{Cite web|url=http:/www.gaelport.com/uploads/documents/edition19.html|title=Eagrán / Edition 19 - 04 11 2008|date=4/11/2008|website=Internet Archive|archive-url=https://web.archive.org/web/20130525011340/http:/www.gaelport.com/uploads/documents/edition19.html|archive-date=25 May 2013|access-date=25 August 2025}}</ref> * ''Táin Bó Cuailnge'', 2018. * ''Bódléar'', 2025. ==== De Bhaldraithe Award ==== The Gradam de Bhaldraithe is awarded to the best work in translation.<ref name=":10" /> * ''Cuairt San Nioclás,'' a translation of Clement Clarke Moore's ''A Visit from St. Nicholas'', or "'Twas the Night Before Christmas."<ref name=":10" /> ==== Other ==== * Walter Macken Prize, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005 * Bháiteir Uí Mhaicín Memorial Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2005<ref>{{Cite news|url=https://www.irishtimes.com/gaeilge/tuarascail/duais-oireachtais-1.501571|title=Oireachtas Prize: Over €50,000 was awarded to writers in the Oireachtas Literary Competitions at an event in Dublin last night. Winners… [Duais Oireachtais: Bronnadh breis agus €50,000 ar scríbhneoirí i gComórtais Liteartha an Oireachtais ar ócáid i mBaile Átha Cliath aréir. Bhuaigh…]|work=5 October 2005|access-date=15 October 2025}}</ref> * BBC Stewart Parker Award, for ''Coinneáil Orainn'' (trans., ''We'll Keep Going''), 2006 * The Aodán Mac Póilín Commemorative Prize, for ''Táin Bó Cuailnge'' (trans., ''The Cattle Raid of Cooley''), 2017 == External Links == * Leabhar Breac website: https://leabharbreac.com/en/ * Leabhar Breac Facebook pages: * Rosg website: [http://www.rosg.ie/en/ <nowiki>http://ww</nowiki>w.rosg.ie/en/] * Art on the Island (Ealaín ar Oileán) website, archived at the Wayback Machine: https://web.archive.org/web/20130601000520/http://ealainaroilean.ie/ 31 March 2012, 1 June 2013 and 8 January 2014 * Darach Ó Scolaí's website Archived 25 September 2015 at the Wayback Machine: https://web.archive.org/web/20150925103456/http://darachoscolai.ie/ * Youtube video of [https://www.youtube.com/watch?v=OlP2AmSBzXc Breandán Ó Cróinin introducing Deirdre at the book launch] in the pub Tigh Mholly (Molly’s House). == Primordial Ooze == * Known for his sensitivity to language and voices. * Finish scanning through JSTOR * Scan through Irish Times, 56 hits * Check Goodreads * Check YouTube (In the spring of 2013, the arts programme Imeall interviewed the author on TG4.) * Check both Wikipedias for pages on the origins of the retold tales (like Deirdre) and link to this article * Propose link from University of Galway page once Darach’s is up * Write Irish National Biography (<nowiki>https://www.dib.ie</nowiki>) to propose an article about Darach once the Wikip article is done? See what they say. * Link to Ó Scolaí from the Wikipedia * Make sure links '''to''' Wikipedia in the actual encyclopedia work right === Not Placed Yet === * "So here are the books that Irish people love the most! [Mar sin seo iad na leabhair is gile leis na Gaeil!]" — "32. An Cléireach – Darach Ó Scolaí (2)" [18 books got 2 votes, and then they're alphabetized by author's last name, so the 32 of 34 doesn't signify the specificity it seems to]<ref name=":14">{{Cite journal|last=Ó Murchú|first=Eoin P.|date=9 June 2017|title=Na 30 leabhar Gaeilge is fearr leis na Gaeil. [The 30 best Irish books for Irish people]|url=https://nos.ie/cultur/leabhair/an-30-leabhar-gaeilge-is-fearr-leis-na-gaeil/|journal=Nós}}</ref> * "Below is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers.Here is a list of those 111 works – a list that shows a great deal of diversity in the reading habits of Irish speakers [Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí.Anseo thíos tá liosta den 111 saothar sin – liosta a léiríonn éagsúlacht an-mhór i nósanna léitheoireachta Gaeilgeoirí]." "Corto Maltese – Hugo Pratt (aistrithe ag Darach Ó Scolaí)"<ref name=":14" /> * "Ceann eile de bhuaicphointí na hÉigse a bheidh sa seisiún le Darach Ó Scolaí, duine d’úrscéalaithe móra na Gaeilge, agus duine de chomhbhunaitheoirí teach foilsitheoireachta Leabhar Breac. [Another highlight of the Éigse will be the session with Darach Ó Scolaí, one of the great Irish language novelists, and one of the co-founders of the publishing house Leabhar Breac.]"<ref name=":17">{{Cite journal|last=Nós|date=4 May 2023|title=Éigse na Bruiséile le filleadh i mí na Bealtaine. [Éigse na Bruséile to return in May]|url=https://nos.ie/cultur/eigse-na-bruiseile-le-filleadh-i-mi-na-bealtaine/|journal=Nós}}</ref> === Things Taken Out for Now === “’The play is a comedy about language, lies, bureaucracy and Gaeltacht grants, in the tradition of Myles na Gcopaleen,’ according to Norma-Jean Kenny in the ''Galway Advertizer'', ‘in which the author comments and criticizes the institutions of the Irish language in Ireland without ceasing.’" Supposedly a quotation by Gearóid Denvir reviewing ''Táin Bó Cuailnge'' for ''Comhar'' (but I don't find it in the article): This book has long been needed by Irish language readers and there is no doubt that it will become a classic in time and surpass Thomas Kinsella’s English version. This version remains faithful to the language of the original while at the same time finding an appropriate language in today’s Irish. Ó Scolaí masterfully overcomes the difficulties of the original’s rhetorical difficulties and the versions of the original poetic texts are extremely effective.[supposedly <ref name=":15" />] “The biggest prize ever awarded for a novel in Irish was presented at a special ceremony in the National Concert Hall in Dublin, today (Thursday, 4 October 2007). Darach Ó Scolaí, writer, artist & playwright from Casla, Co. Galway, was awarded €10,000 for his literary novel, ‘An Ardscoil’. This work, under the new title ‘An Cléireach’, will be launched at Oireachtas na Samhna in Westport in November. This is the first novel from his pen, a story set in the late seventeenth century. This competition was sponsored by the Franciscan Province of Ireland.” (archive, Oireachtas na Gaeilge site, 04 October, 2007) ''Súil an Daill'' (trans., ''The Eye of the Blind''), Leabhar Breac, 2021. number 2 in ''Comhar'' literary magazine’s list of best books of 2021. '''{6}.''' *William Shakespeare, ''Romeo agus Juliet'' (trans. of ''Romeo and Juliet''), Leabhar Breac, 2016. *Jonathan Swift, ''Camchuairt Ghuilivéir'' (trans. of ''Gulliver's Travels''), Leabhar Breac, 2016. *Hugo Pratt, ''Corto Maltese'' '''''Flag of Bones (Bratach na gCnámh) Series''''' Leabhar Breac published the Bratach na gCnámh series of books for young readers. Written in French by Alain Surget, illustrated by Annette Marnat and translated by Darach Ó Scolaí, this series uses the history of Caribbean Sea pirates<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/alain-surget/|title=Alain Surget Archives|website=Leabhar Breac|language=en-US|access-date=2025-09-30}}</ref>: *Alain Surget, ''Éalú as Páras'' (''Escape from Paris''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Oilean na Siorcanna'' (''Shark Island''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''Long na dTaibhsi'' (''Ship of the Ghosts''), Annette Marnat (Illustr.), Leabhar Breac, 2011. * Alain Surget, ''San Ochtapas Dubh'' (''In the Black Octopus''), Annette Marnat (Illustr.), Leabhar Breac, 2013. * Alain Surget, ''San Ionsai ar Veracruz'' (''The Attack on Veracruz''), Annette Marnat (Illustr.), Leabhar Breac, 2013. '''''For "First Readers" (children to 6 years old or so)''''' These books were written originally in Catalan by Spanish author Enric Lluch Girbés and translated into Irish by Ó ScolaÍ: *Enric Lluch, ''Ag Péinteáil an Tí'' (''Painting the House''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Colúr Bacach'' (''The Lazy Dove''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''An Phluais'' (''The Cave''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Madra Dhaideo'' (''Grandpa's Dog''), Anna Clariana (Illustr.), Leabhar Breac, 2017. *Enric Lluch, ''Fiacail Mháire'' (''Mary's Tooth''), Anna Clariana (Illustr.), Leabhar Breac, 2017. '''''Bruno Heitz''''' Leabhar Breac published a series of 3 Heitz books for small children. Published originally in French, this series of three comic books is about a blind mole named Cáitín Chaoch in Irish (and ''Louisette la taupe'' in French).<ref>{{Cite web|url=https://leabharbreac.com/en/product-category/bruno-heitz-en/|title=Bruno Heitz Archives|website=Leabhar Breac|language=en-US|access-date=2025-10-02}}</ref> Ó Scolaí translated these: *Bruno Heitz (author and illustr.), ''Práinneach'' (''Urgent''), Leabhar Breac, 2020 *Bruno Heitz (author and illustr.), ''Preab san Aer'' (''Bounce in the Air''), Leabhar Breac, 2020. '''''Books for Toddlers''''' Leabhar Breac has published 14 books written by French author Orianne Lallemand's and illustrated by Eleonore Thuillier, about Lallemmand's popular character Loup, Wolf. These are translated by Ó Scolaí: *Orianne Lallemand, ''An Mac Tire a Raibh Faitios an Domhain Air'' (trans. of ''The Son Who Saw the World in His Eyes''), Eleonore Thuillier  Illustr.), Leabhar Breac, 2018. *Orianne Lallemand, ''Macan agus an Goban'' (trans. of ''Macan and the Goblin''), Eleonore Thuillier (Illustr.), Leabhar Breac, 2018. * Orianne Lallemand, ''A Mac Tíre a Chuaigh go Tóin na Farraige'' (trans. of ''The Wolf Who Went to the Bottom of the Sea''), Éléanore Thuillier  (Illustr.), Leabhar Breac, 2019. '''''Board Books (for babies)''''' J. C. (Joan Carles) Girbés Aparisi is a Catalan author and editor. These books were written in Catalan and translated by Ó Scolai. *J. C. Girbés, ''An Phicnic'' (''The Picnic''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. * J. C. Girbés, ''An Chóisir'' (''The Party''), Silvia Ortega (Illustr.), Leabhar Breac, 2013. *J. C. Girbés, ''Lá Mór Fada'' (''A Long Day''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. *J. C. Girbés, ''Tabhair Leat do Leabhar'' (''Bring Your Book''), Silvia Ortega (Illustr.), Leabhar Breac, 2014. ==== Gradam Réics Carló ==== The Réics Carló prize is awarded for the best book in the Irish language for young readers. It is named for one of the characters of 20th-century writer [[wikipedia:Cathal_Ó_Sándair|Cathal Ó Sándair (Charles Saunders)]]. * ''An Bradán Feasa'' was “shortlisted for the Réics Carlo award 2010.”<ref name=":9" /> == References == {{reflist}} 93thfen3058xno4vca4w7w36cokw1vp 3 274033 2835529 2831227 2026-10-02T17:11:15Z MediaWiki message delivery 983498 /* Wikimedians for Sustainable Development - September 2026 Newsletter */ new section 2835529 wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] IanVG!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOUI JS signature icon LTR.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:06, 27 April 2021 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == Thank you Dave! == Well, I am obviously replying far too late, but thank you @[[User:Dave Braunschweig|Dave Braunschweig]] for your kind words. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:25, 29 July 2025 (UTC) == Possible copyright problem == FYI, I just tagged [[Introduction to thermodynamics/Extensive and intensive properties]] for a possible copyright problem: CC NonCommercial not compatible with CC-BY-SA. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:41, 8 November 2025 (UTC) :Hey @[[User:Dan Polansky|Dan Polansky]]! Thanks a ton, I didn't know that the Non-Commercial part of the license their work was licensed under led to that kind of copyright issue! Thanks for education and help :-). I haven't yet reached out, but I'm going to look for a format email that maybe someone else has already written on the wikipedia to help with this. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:27, 12 November 2025 (UTC) ::Okay, just sent an email it looks something like: ::Dear Dr. Claire Yu Yan, ::I wanted to let you know that I enjoyed your Introduction to Engineering Thermodynamics at https://pressbooks.bccampus.ca/thermo1/, which I found while researching for the free, online and open learning resources on "Wikiversity" (sister project of "Wikipedia"); I thought that your information on the subject is worthy of inclusion in our living and growing document. ::Wikiversity ( ::https://www.wikiversity.org ::) is a Wikimedia Foundation project that is collaboratively edited by volunteers from around the world. Our goal is to create and host a living, and diverse set of learning resources, learning projects and research for use by teachers, students and researchers. It is not only available at no charge, but also freely distributed. It is one of many projects of the non-profit Wikimedia Foundation. ::We can only use your material if you are willing to grant permission for it to be used under terms of the Creative Commons Attribution-ShareAlike 4.0 International License. This means that although you retain the copyright and authorship of your own work, you are granting permission for all others (not just Wikipedia) to use, copy, and share your materials freely—and even potentially use them commercially—as long as they do not try to claim the copyright themselves, or try to prevent others from using or copying them freely. You can read this license in full at:   ::(https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License) ::Please note that your contributions may not remain intact as submitted; this license, and the collaborative nature of our project, also entitles others to edit, alter, and update them at will, i.e., to keep up with new information, or suit the text to a different purpose. However, the license also expressly protects authors "from being considered responsible for modifications made by others" while ensuring that those authors get credit for their work. There is more information on our copyright policy at:   ::(https://en.wikipedia.org/wiki/Wikipedia:Copyrights) ::We choose the Creative Commons Attribution-ShareAlike 4.0 International License because it is the best available tool for ensuring that our encyclopedia is and can remain free for all to use, and for providing credit to everyone who donates text and images. It may or may not be compatible with your goals in creating the materials available on your website — that is your choice. Please be assured that if you do not grant permission, your materials will **not** be used at Wikipedia; we have a very strict policy against copyright violations. ::If you do agree to grant permission, we will credit you for your work in the resulting article's references section, by stating it was based on your work and is used with your permission, and by providing a link back to your website. ::You are obviously an expert in your field, and we invite your active collaboration in writing and editing articles on this subject and any others that might be attractive to you. If you are interested, please see:   ::([[Wikiversity:Introduction|https://en.wikiversity.org/wiki/Wikiversity:Introduction]]) for more information! ::Thank you for your time; we look forward to your reply. ::Kindly,   ::<My name> [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:48, 12 November 2025 (UTC) :::Have you heard back about the licensing issue? [[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:35, 17 August 2026 (UTC) ::::I unfortunately did not... :(. I'll start removing the content later this week. Should I move them to a sandbox type area, or just delete it outright? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 15:21, 18 August 2026 (UTC) == Wikimedians for Sustainable Development - January 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty first newsletter. This issue has news related to SDG 15.<div style="column-count:2; column-width: 400px;"> ; User group news * The [[m:Wikimedians for Sustainable Development/Reports/2025|annual report for 2025]] was published. * [[m:Wikimedians for Sustainable Development/Next meeting|Next user group meeting]] is 22 February. * The drafting of the [[m:Wikimedians for Sustainable Development/Annual plan 2026|2026 annual plan]] is under way, please help. ; Other news * [https://diff.wikimedia.org/2026/01/09/winning-images-of-the-special-category-human-rights-and-environment-from-wiki-loves-earth-2025%F0%9F%A4%9D/ Winning images of the special category “Human Rights and Environment” from Wiki Loves Earth 2025🤝] (SDG 15) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 14:22, 4 February 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=29928029 --> == Wikimedians for Sustainable Development - February 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty second newsletter. This issue has news related to SDG 4, 5, 10, 13, 15, 16 and 17.<div style="column-count:2; column-width: 400px;"> ; User group news * A proposal for a climate and sustainability meetup at Wikimania has been submitted. Keep your fingers crossed it gets accepted! ; Other news * [https://metabase.wikibase.cloud Metabase], a project to create a [[m:Movement Strategy/Initiatives/Knowledge Base|movement-wide knowledgebase for activities and initiatives]], now has the property [https://metabase.wikibase.cloud/wiki/Property:P109 relates to sustainable development goal, target or indicator] and all the Sustainable Development Goals, Targets and Indicators. This makes it possible to make sure that your projects and initiative that supports these are marked as doing so and also find previous efforts related to them. * [https://diff.wikimedia.org/2026/02/11/wiki-for-botanists-why-thematic-engagement-matters/ Wiki for Botanists: Why thematic engagement matters] (SDG 15) * [https://diff.wikimedia.org/2026/02/15/influence-of-seasonal-and-eco-climatic-factors-on-butterfly-diversity-insights-from-wiki-loves-butterfly/ Influence of Seasonal and Eco-climatic Factors on Butterfly Diversity: Insights from Wiki Loves Butterfly] (SDG 15) * [https://diff.wikimedia.org/2026/02/15/african-women-in-climate-action-a-continued-editing-journey-through-the-edither-africa-contest-2026/ African Women in Climate Action: A Continued Editing Journey through the EditHer Africa Contest 2026] (SDG 5 & 13) ; Events * March is Women's History Month and also has the Internaltional Women's day, so there are plenty of related events. Check out [[m:Special:AllEvents|Special:AllEvents]] to find some near you. (SDG 5) * [[m:Wiki Loves Ramadan 2026|Wiki Loves Ramadan 2026]] (SDG 16) * [[d:Wikidata:WikiProject_India/Events/International_Mother_Language_Day_2026_Datathon|International Mother Language Day 2026 Datathon]] (SDG 4, 10 &17) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 12:13, 2 March 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=29928029 --> == Wikimedians for Sustainable Development - March 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty third newsletter. This issue has news related to SDG 15.<div style="column-count:2; column-width: 400px;"> ; User group news * There is now a [[c:Template:User Wikimedians for Sustainable Development|user box template on Wikimedia Commons]] that you can use to show that you are participant of the user group. There were already user boxes on [[m:Template:User Wikimedians for Sustainable Development|Meta]], [[d:Template:User Wikimedians for Sustainable Development|Wikidata]], [[w:en:Template:User Wikimedians for Sustainable Development|English]] and [[w:sv:Mall:Användare Wikimedians for Sustainable Development|Swedish]] Wikipedia. If your home wiki uses user boxes but lacks one, feel free to copy any of these to it. ; Other news * [https://wikimediafoundation.org/news/2026/03/02/the-winners-of-wiki-loves-earth-2025/ “Cinematic intensity”: The winners of Wiki Loves Earth 2025] (SDG 15) * [https://www.nature.com/articles/d41586-026-00940-y Scientists should join collaborative online editing communities for biodiversity] (SDG 15) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 11:26, 1 April 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30155800 --> == Wikimedians for Sustainable Development - April 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty fourth newsletter. This issue has news related to SDG 2, 5, 6, 7, 13 and 15.<div style="column-count:2; column-width: 400px;"> ; News * [[diffblog:2026/04/15/from-lens-to-knowledge-citizen-science-through-wiki-loves-butterfly/|From Lens to Knowledge: Citizen Science through Wiki Loves Butterfly]] (SDG 15) * [[outreach:GLAM/Newsletter/March 2026/Contents/Biodiversity Heritage Library report|Wikidata type specimen data model]] (SDG 15) * [[outreach:GLAM/Newsletter/March 2026/Contents/Macedonia report|Edit-a-thon "Women Botanists" and “Plants Around Us: Veles” workshop]] (SDG 5 & 15) ; Events * Ongoing: [[w:en:Wikipedia:100 Days 100 Edits|100 Days 100 Edits]] (SDG 13) * Ongoing: [[m:Wiki for Sustainable Futures 2026|Wiki for Sustainable Futures 2026]] (SDG 2, 6 & 7) * May 9-10: [[w:sv:Wikipedia:Projekt naturgeografi/Fotosafari: Fåglar i Skåne 2026|Bird photography trip]] in south Sweden (SDG 15) * May 30: [[w:sv:Wikipedia:Skrivstuga/Biologisk mångfald|Editathon about biodiversity]] in Stockholm (SDG 15) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 12:47, 6 May 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30472224 --> == Wikimedians for Sustainable Development - May 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty fifth newsletter. This issue has news related to SDG 2, 6, 7, 13 and 15.<div style="column-count:2; column-width: 400px;"> ; News * [[diffblog:2026/05/19/wikimedia-projects-and-the-climate-crisis-how-wiki-for-sustainable-futures-2026-is-being-built/|Wikimedia Projects and the Climate Crisis: How Wiki for Sustainable Futures 2026 Is Being BuiltWikimedia Projects and the Climate Crisis: How Wiki for Sustainable Futures 2026 Is Being Built]] (SDG 2 & 6 & 7 & 13) * [https://wikiedu.org/blog/2026/05/21/earth-day-every-day-preserving-biodiversity-on-wikipedia/ Earth Day, Every Day: Preserving Biodiversity on Wikipedia] (SDG 15) *[[diffblog:ar/2026/05/29/%d8%a7%d9%86%d8%b7%d9%84%d8%a7%d9%82-%d9%85%d8%b3%d8%a7%d8%a8%d9%82%d8%a9-%d8%a7%d9%84%d8%a8%d9%8a%d8%a6%d8%a9-%d8%a7%d9%84%d8%b9%d8%b1%d8%a8%d9%8a%d8%a9-2026-%d9%85%d8%a8%d8%a7%d8%af%d8%b1%d8%a9/|Launch of the Arabic Environmental Contest 2026: an ambitious initiative to enrich environmental content]] (SDG 2 & 6 & 7) * [https://wikimedia.org.au/wiki/From_the_field_to_the_free_web From the field to the free web] (SDG 15) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 17:47, 1 June 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30606764 --> == Wikimedians for Sustainable Development - June 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty sixth newsletter. This issue has news related to SDG 4, 5 and 10.<div style="column-count:2; column-width: 400px;"> ; In the news * [[diffblog:2026/06/08/reading-wikipedia-in-the-classroom-in-wa-ghana-empowering-educators-with-media-and-information-literacy-skills/|Reading Wikipedia in the Classroom in Wa, Ghana: Empowering Educators with Media and Information Literacy Skills]] (SDG 4) * [[diffblog:2026/06/12/amplifying-womens-stories-and-indigenous-knowledge-feminism-and-folklore-2026-in-the-igbo-community/|Amplifying Women’s Stories and Indigenous Knowledge: Feminism and Folklore 2026 in the Igbo Community]] (SDG 5) * [[diffblog:2026/06/13/artfeminism-network-organizers-at-eseap-conference-2026/|Art+Feminism Network Organizers at ESEAP Conference 2026]] (SDG 5) * [[diffblog:2026/06/14/a-reflection-on-what-i-learned-at-my-first-international-womens-day-celebration/|A Reflection on What I Learned at My First International Women’s Day Celebration]] (SDG 5) * [[diffblog:2026/06/16/from-mentee-to-builder-my-six-months-in-the-eduwiki-hub-mentorship-program/|From Mentee to Builder: My Six Months in the EduWiki Hub Mentorship Program]] (SDG 4) * [[diffblog:2026/06/16/building-skills-and-confidence-during-my-three-month-journey-through-the-on-wiki-skill-program-organized-by-africa-wiki-women/|Building Skills and Confidence During My Three-Month Journey Through the On Wiki Skill Program Organized by Africa Wiki Women]] (SDG 5) * [[diffblog:2026/06/17/why-the-eduwiki-starter-kit-matters-for-the-future-of-education/|Why the EduWiki Starter Kit Matters for the Future of Education]] (SDG 4) * [[diffblog:2026/06/17/empowering-new-editors-to-bridge-the-gender-gap-my-experience-as-a-mentor-in-the-edither-africa-april-2026-contest/|Empowering New Editors to Bridge the Gender Gap: My Experience as a Mentor in the EditHer Africa April 2026 Contest]] (SDG 5) * [[diffblog:2026/06/18/visible-women-edit-a-thon-piloting-a-collaborative-approach-to-free-knowledge-in-hong-kong/|Visible Women Edit-a-thon: Piloting a Collaborative Approach to Free Knowledge in Hong Kong]] (SDG 5) * [[diffblog:2026/06/23/guiding-new-voices-training-women-in-wikidata-during-the-april-edither-africa-contest/|Guiding New Voices: Training Women in Wikidata during the April EditHer Africa Contest]] (SDG 5) * [[diffblog:2026/06/23/wikimedia-community-usergroup-botswana-in-collaboration-with-art-and-feminism-on-wikidata-mobile-training-2026/|Wikimedia Community Usergroup Botswana in collaboration with Art and Feminism on Wikidata Mobile Training 2026]] (SDG 5) ; Events * 10-18 July: [[m:Event:Wikiwomen Camp 2026|Wikiwomen Camp 2026]] (SDG 5) * 14 July [[w:de:Veranstaltung:Queers & Frauen 14.07.2026|Queers & Frauen]] (SDG 5 & 10) * 21-25 July: Wikimania is coming up and there are plenty of sessions related to sustainable development. There is now a dedicated page for those sessions at [[wikimania:2026:Program/SDG related sessions]]. * 23 July: [[m:Event:From Constitution to Community — Documenting Queer Rights Movements|From Constitution to Community — Documenting Queer Rights Movements]] (SDG 10) * 14 July [[w:de:Veranstaltung:Queers & Frauen 28.07.2026|Queers & Frauen]] (SDG 5 & 10) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 19:20, 3 July 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30633243 --> == Wikimedians for Sustainable Development - July 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty seventh newsletter. This issue has news related to SDG 5, 6, 15 and 16.<div style="column-count:2; column-width: 400px;"> ; News * [[diffblog:2026/07/04/photowalk-at-the-purbachal-reserve-forest/|Photowalk at the Purbachal Reserve Forest]] (SDG 15) * [[diffblog:2026/07/13/welcoming-women-into-wikimedia-tech-a-guide-based-on-lived-experiences/|Welcoming Women+ into Wikimedia Tech: A Guide Based on Lived Experiences]] (SDG 5) * [[diffblog:2026/07/14/empowering-youth-through-knowledge-wikipedia-for-freedom-of-elections-in-armenia/|Empowering Youth Through Knowledge: “Wikipedia for Freedom of Elections” in Armenia]] (SDG 16) * [[diffblog:2026/07/19/water-for-life-knowledge-for-all-how-wikiverse-botswana-is-showcasing-africas-water-story-through-the-africa-wiki-challenge-2026/|Water for Life, Knowledge for All: How WikiVerse Botswana is Showcasing Africa’s Water Story Through the Africa Wiki Challenge 2026]] (SDG 6) * Wikimania: There were so many sessions and posters related to the SDGs, they can't all be listed here. But on [[wikimania:2026:Program/SDG related sessions|this Wikimania page]] you can find them all and the session pages have links to the recordings. This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 08:56, 4 August 2026 (UTC)• [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30853204 --> == Wikimedians for Sustainable Development - August 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty eighth newsletter. This issue has news related to SDG 3, 4, 10, 11, 13 and 14.<div style="column-count:2; column-width: 400px;"> ; News * [[diffblog:2026/08/12/wiki-loves-fish-bringing-coastal-biodiversity-to-the-world-through-open-knowledge/|Wiki Loves Fish: Bringing Coastal Biodiversity to the World Through Open Knowledge]] (SDG 14) * [https://wikiedu.org/blog/2026/08/19/making-rare-disease-knowledge-a-little-more-common-on-wikipedia/ Making rare disease knowledge a little more common on Wikipedia] (SDG 3) * [[diffblog:2026/08/26/the-pre-conference-on-information-integrity-in-climate-change-at-wikimania/|The Pre-Conference on Information Integrity in Climate Change at Wikimania]] (SDG 13) * [[diffblog:2026/08/29/beyond-wikipedia-a-technical-workshop-introducing-moroccan-teachers-to-wikidata-and-the-wikimedia-technical-ecosystem/|Beyond Wikipedia: A Technical Workshop Introducing Moroccan Teachers to Wikidata and the Wikimedia Technical Ecosystem]] (SDG&nbsp;4) * [[diffblog:2026/08/29/wiki-for-human-rights-nigeria-write-for-rights-and-july-virtual-session/|Wiki for Human Rights Nigeria: Write for Rights and July Virtual Session]] (SDG 10) ; Events * Ongoing: [[c:Commons:Wiki Loves Monuments 2026|Wiki Loves Monuments]] (SDG 11) This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 08:21, 4 September 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=30910094 --> == Wikimedians for Sustainable Development - September 2026 Newsletter == <div lang="en" dir="ltr" class="mw-content-ltr">This is our fifty ninth newsletter. This issue has news related to SDG 4, 5, 11, and 15.<div style="column-count:2; column-width: 400px;"> ; News * [https://wikimedia.org.au/wiki/The_little_coloured_bar_that_tells_you_a_species_is_disappearing_%E2%80%94_and_the_Australian_project_keeping_it_accurate The little coloured bar that tells you a species is disappearing — and the Australian project keeping it accurate] (SDG 15) * [[diffblog:2026/09/05/more-than-a-campaign-the-measurable-impact-of-celebrate-women-2026/|More than a campaign: the measurable impact of Celebrate Women 2026]] (SDG 5) * [[diffblog:2026/09/04/shesaid-2026-the-global-story-continues-one-voice-at-a-time/|SheSaid 2026: The Global Story Continues, One Voice at a Time]] (SDG 5) * [[diffblog:2026/09/13/from-focus-group-to-skills-focus-a-new-programme-of-wiki-loves-women/|From Focus Group to Skills Focus: A New Programme of Wiki Loves Women]] (SDG 5) * [[diffblog:2026/09/17/2387-photographs-237-participants-documenting-colombias-biodiversity-through-wiki-loves-earth-2026/|2,387 photographs, 237 participants: documenting Colombia’s biodiversity through Wiki Loves Earth 2026]] (SDG 15) * [[diffblog:2026/09/20/more-programs-arent-enough-why-eduwiki-needs-stronger-infrastructure/|More Programs Aren’t Enough: Why EduWiki Needs Stronger Infrastructure]] (SDG 4) * [[diffblog:2026/09/21/wiki-loves-earth-2026-in-bangladesh-3330-photographs-15-winners-and-one-snake-that-changed-the-rules/|Wiki Loves Earth 2026 in Bangladesh: 3,330 photographs, 15 winners, and one snake that changed the rules]] (SDG 15) * [[diffblog:2026/09/23/from-open-knowledge-to-national-development-representing-wikimedia-botswana-at-the-un-consultation/|From Open Knowledge to National Development: Representing Wikimedia Botswana at the UN Consultation]] * [[diffblog:2026/09/25/from-university-research-to-free-knowledge-academics-discover-wikimedia/|From University Research to Free Knowledge: Academics Discover Wikimedia]] (SDG 4) * [[diffblog:2026/09/26/after-the-storm-the-shoots-are-more-photogenic/|After the storm, the shoots are more photogenic]] (SDG 15) ; Events * Still ongoing: [[c:Commons:Wiki Loves Monuments 2026|Wiki Loves Monuments]] (SDG 11) </div> This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by <bdi lang="en" dir="ltr">[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])</bdi> 17:11, 2 October 2026 (UTC) • [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]] </div> <!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&oldid=31035794 --> ghr60f5ke3vff72qm3rg5zwyzagqwhl 2 283170 2835643 2834497 2026-10-03T07:53:38Z Elominius 2911372 + trash-cli 2835643 wikitext text/x-wiki ; Possibly useful packages for Linux users: 7z mediainfo ffmpeg mplayer mpv mkvtoolnix sdparm udftools qpxtool dvd+rw-tools cdrkit cdrtools xorriso hwinfo lsscsi gameconqueror xdotool autokey blender inkscape iotop htop filezilla k3b lmms pavucontrol virtualbox sqlite3 tlp bluez blueberry kdenlive picard easytag libimage-exiftool-perl vsftpd proftpd-basic nemo file-roller searchmonkey catfish lzip lziprecover xed kate hexedit ghex ddrescue foremost sleuthkit testdisk gparted git perf-tools xvattr imap-tools android-tools-adb sg3-utils libimage-exiftool-perl ucommon-utils smartmontools blktrace trash-cli 57q7gj4n4vgpwaf5nfmwi3ijt6g0mha 0 285380 2835609 2835328 2026-10-03T04:49:20Z ~2026-53146-27 3112023 /* Applications */ 2835609 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.20261002.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> 56yqd9bxbpwrlfch477bmfekmkqelf0 4 285491 2835540 2834397 2026-10-02T17:32:22Z Alexis Jazz 791434 Updating gadget usage statistics from [[Special:GadgetUsage]] ([[phab:T121049]]) 2835540 wikitext text/x-wiki {{#ifexist:Project:GUS2Wiki/top|{{/top}}|This page provides a historical record of [[Special:GadgetUsage]] through its page history. To get the data in CSV format, see wikitext. To customize this message or add categories, create [[/top]].}} The following data is cached, and was last updated 2026-10-01T06:14:32Z. A maximum of {{PLURAL:5000|one result is|5000 results are}} available in the cache. {| class="sortable wikitable" ! Gadget !! data-sort-type="number" | Number of users !! data-sort-type="number" | Active users |- |CleanDeletions || 75 || 0 |- |EnhancedTalk || 1402 || 7 |- |HideFundraisingNotice || 841 || 7 |- |HotCat || 922 || 10 |- |LintHint || 119 || 2 |- |Round Corners || 1190 || 3 |- |contribsrange || 388 || 3 |- |dark-mode || 135 || 3 |- |dark-mode-toggle || 189 || 5 |- |edittop || 520 || 7 |- |popups || 884 || 5 |- |purge || 738 || 12 |- |sidebartranslate || 561 || 1 |- |usurper-count || 119 || 1 |} * [[Special:GadgetUsage]] * [[m:Meta:GUS2Wiki/Script|GUS2Wiki]] <!-- data in CSV format: CleanDeletions,75,0 EnhancedTalk,1402,7 HideFundraisingNotice,841,7 HotCat,922,10 LintHint,119,2 Round Corners,1190,3 contribsrange,388,3 dark-mode,135,3 dark-mode-toggle,189,5 edittop,520,7 popups,884,5 purge,738,12 sidebartranslate,561,1 usurper-count,119,1 --> 4kt98ebmr2skya3oddabqp6l85g3a38 2 289273 2835547 2835267 2026-10-02T18:24:07Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835547 wikitext text/x-wiki {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|June 2023 - September 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the Kepler problem of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain 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. 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 0jba6fyaqccztywg41p709jg22xxas4 2835548 2835547 2026-10-02T18:24:25Z Dc.samizdat 2856930 2835548 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the Kepler problem of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain 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. 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 3msmg27bbczp9zc4omuacwc5webmlxo 2835549 2835548 2026-10-02T18:27:12Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835549 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. Only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain 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. 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} ag6n43sju0boj3ovji9z5q3jx2xd3zg 2835550 2835549 2026-10-02T18:30:03Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835550 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain 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. 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 4mn3lnkljdbsyrmzq45a1kblx0ec80u 2835558 2835550 2026-10-02T19:43:36Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835558 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand screw orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi</math>, but they are true geodesic circles. We shall refer to such circles in 4-space as ''isoclines'', to distinguish them from ordinary great circles in the plane. The name comes from the fact that a point on a rigid object undergoing 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 3suh05ymh1smdz8by2zrjehy5pl5iv2 2835559 2835558 2026-10-02T19:44:23Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835559 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand screw orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi</math>, but they are true geodesic circles. We shall refer to such circles in 4-space as ''isoclines'', to distinguish them from ordinary great circles in the plane. The name comes from the fact that a point on a rigid 4-dimensional object undergoing 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} oarv8rh54guc4m8hh8wpom0gcs0sg92 2835566 2835559 2026-10-02T20:34:21Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835566 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We shall refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 1ld8413rv0w7ks4vvb4o5skb8uzpibp 2835567 2835566 2026-10-02T20:34:58Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835567 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} ex01pf7ocpecfqq1sydrt9v0h73wq31 2835568 2835567 2026-10-02T20:38:22Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835568 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 7uj00rct2ca77w287epcph3adapeg8w 2835569 2835568 2026-10-02T20:47:05Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2835569 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} jkf0h6ukpeqdw39etgwvk5qyjq3q74w 2835570 2835569 2026-10-02T20:50:54Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2835570 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double-the-unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} ky4mxy2lwv5cezt86h9ha0f6blm028h 2835571 2835570 2026-10-02T20:51:51Z Dc.samizdat 2856930 /* Light propagates through 4-space at twice its apparent velocity c */ 2835571 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional 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 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} m4yr01b28s5h5a3rm388rn7jf5yqo5k 2835581 2835571 2026-10-02T22:41:07Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835581 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} bayeo5kf4smj4vfpub3e29ew8bnjods 2835596 2835581 2026-10-03T01:12:01Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835596 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space will contain no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} b2mznb8pjb6a5uxrk0v32vqhb6cfrmp 2835597 2835596 2026-10-03T01:29:20Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835597 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand orientation. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} b35sw9ks0ak2iu5o8rh6mel9ouqwezn 2835598 2835597 2026-10-03T01:32:55Z Dc.samizdat 2856930 /* An object's motion in space is the product of its discrete self-reflections */ 2835598 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} 22ocq96r90x8n81dgd18w12pm36s2zb 2835599 2835598 2026-10-03T01:40:11Z Dc.samizdat 2856930 /* Special relativity describes Euclidean 4-space */ 2835599 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} * {{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}} fvxjmfj46mii2zdtj4fc4kc67pxfssm 2835600 2835599 2026-10-03T01:44:38Z Dc.samizdat 2856930 2835600 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 our planet are only 3-dimensional objects, but nonetheless we can see in four dimensions of space. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. Light from them reaches us directly, on straight lines through 4-space. This view of the observed universe is compatible with special and general relativity, and with quantum mechanics. It furnishes those theories with an explanatory geometric model.</blockquote> == Summary == We observe that: * Physical space has four perpendicular dimensions, not just three. * Atoms are [[W:4-polytope|4-polytopes]]. * The sun is a [[W:4-ball|4-ball]] that is round in four dimensions. * Everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space. * Our entire 3-space manifold is moving through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions. * Special relativity describes the linear motion of objects and light signals in four-dimensional Euclidean space. == A theory of the Euclidean cosmos == The physical universe is properly visualized as [[w:Four-dimensional_space|real Euclidean space <math>\mathbb{R}^4</math>]] of four orthogonal spatial dimensions. Space itself has a fourth perpendicular dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are [[W:4-ball|4-balls]] of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, are so flat as to be essentially 3-dimensional, having only the thickness of an atom in the orthogonal fourth dimension. All objects with mass move inertially through Euclidean 4-space at constant velocity <math>c</math> as long as they exist, and acceleration only varies their direction. Objects moving in the same direction are in the same inertial reference frame. Their direction of motion through 4-space at velocity <math>c</math> is their proper time dimension, simply because their direction and velocity of motion through time is the same as their direction and velocity of motion through space. A typical galaxy such as ours occupies a 4-ball of mostly empty space, with stars and other objects distributed non-uniformly within it. The galaxy's orbital center may be nothing: a smaller 4-ball of empty space they surround. The stars in our spiral galaxy appear from our accustomed point of view to occupy a disk-like region of 3-dimensional space, with a denser ball-like center, but they are not so confined: they are distributed within a region of 4-dimensional space. The disk and ball have thickness not only in the third dimension, but in the fourth spatial dimension as well. Light from stars and galaxies reaches us on straight lines through Euclidean 4-space, so from our viewpoint, in what we have always imagined to be a 3-space universe, we look into the surrounding 4-space. Although we are physically confined within a 3-dimensional hyperplane by the inertia of our motion through 4-space at constant velocity <math>c</math>, light signals are not so confined, so we observe all the cosmological objects surrounding us, both in and above or below our hyperplane, without perceiving their separation in our fourth orthogonal dimension, the direction of our motion. We may perceive a galaxy as elliptical when it is actually spherical, because that is how its image projects from its 4-ball shape into a 3-ball region in our hyperplane, where we measure it with our 3-dimensional bodies. More generally, orbits are circular in 4-space, but elliptical in the 3-space of their elliptic hyperplane. The galaxy as a whole, or more properly its orbital barycenter, is translating through 4-space at velocity <math>c</math>, in a distinct direction orthogonal to all three dimensions of our ordinary proper 3-space. Stars within the galaxy are translating with it at the same velocity <math>c</math> in the same direction, but on spiral trajectories as they pursue their various orbits within the galaxy. The galaxy as a whole occupies a 4-ball within its proper inertial reference frame (that is, in the moving frame of reference in which the galaxy considers itself to be a stationary rotating 4-ball). Over time, the galaxy 4-ball occupies a 4-dimensional cylinder and progresses along the cylinder's axis at velocity <math>c</math>. In this more universal inertial reference frame, the stars in the galaxy follow helical geodesic paths through the 4-cylinder; their trajectories are screw-translations through 4-space, the compound of a simple rotation and a completely orthogonal linear translation. The gravitational force and the inertial tendency to follow a geodesic are the same phenomenon, by the equivalence principle. That said, they can be distinguished, and the galaxy is held together primarily by gravity as inertia, not by gravity as attraction to a central mass toward which objects fall in orbit. There is not enough mass in the galaxy to hold it together by attraction, there is just enough to bend the stars' trajectories toward each other, in helical orbits around a barycentric axis of motion. It is the tremendous inertial force of stars in translational motion at velocity <math>c</math> that holds the cylinder of motion together, not some invisible dark matter. The observed universe as a whole appears to be a [[W:3-sphere|3-sphere]] expanding radially from a central origin point at velocity <math>c</math>, the invariant velocity of mass-carrying objects through 4-space, also the propagation speed of light relative to any moving 3-space manifold, as measured by all observers. A 3-sphere is a hypersphere in Euclidean 4-space, but the enclosing surface of a 3-sphere is a curved, finite 3-space, dimensionally analogous to the surface of the earth which is a curved, finite 2-space. This 3-sphere could be the domain of our visible cosmos, but of course we do not know for certain that all the cosmological objects we observe lie near the surface of our expanding 3-sphere, since it is only our assumption that they must all have originated in the same big bang long ago. Possibly some of the more distant objects we observe did not, and lie elsewhere, outside our big-bang's 3-sphere of outflying matter or even inside its 3-sphere, below its surface. We cannot assume that all objects in the 4-space universe lie near the surface of the same expanding 3-sphere. For all observers, the conjectured big-bang of their origin corresponds not only to a now-distant point in their proper time past, it also corresponds to a distinct now-distant point in 4-dimensional space: the same point in the same Euclidean 4-space for all observers with the same origin. Our big bang had a distinct origin point in real space as well as in real time. More generally, time and Euclidean 4-space can be measured independently, just as time and Euclidean 3-space were measured classically, without the necessity to combine them as spacetime. The same inertial force which holds the galactic cylinder of motion together also confines us physically to an exceedingly thin three-dimensional surface manifold moving through 4-space at velocity <math>c</math>. All objects in our solar system except the sun itself lie within this thinest three-dimensional manifold, and have only the thickness of an atom in their direction-of-motion fourth dimension. That is why we are 3-dimensional objects ourselves, and why we cannot construct more than three perpendiculars through a single point in our local 3-dimensional space. A spherical region of 4-space is called a [[W:4-ball|4-ball]]. The enclosing surface of a 4-ball of any size is a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. Our entire big-bang-origin universe appears to be the largest 3-sphere we observe, but each of the cosmological objects within it (including our galaxy and our sun) is contained in a smaller 3-sphere shell of its own, 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, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the surface of a dimensionally rounder space, although that surface is not infinite. It is curved, and finite, analogous to the way the 2-dimensional surface of the earth -- once thought to be flat -- is curved and finite. Our 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 interior of stars and supermassive objects. Our 3-manifold of ordinary space, as a surface within our moving 4-ball galaxy, is translating through 4-space at velocity <math>c</math> with the galaxy, in a distinct direction orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the translation through 4-space is following a geometric law of motion discovered by Coxeter, that governs the propagation of individual objects through Euclidean space by the actions of their symmetry groups. The solar system's atoms of mass are 4-polytopes that are simultaneously rotating and translating, and as they advance together they define a moving 3-dimensional manifold by their own collective inertia, also called gravity, the property of matter's ceaseless propagation through 4-space at the constant velocity <math>c</math>, the universal rate of causality at which quantum events occur, all objects move, and the universe evolves. Any moving 3-dimensional manifold such as ours is an evolving surface boundary that is empty in most places, occupied by single atoms in comparatively fewer places, and occupied by bound complexes of multiple atoms (molecules) in still fewer places. In all these places it is no thicker than one atom in the dimension corresponding to its direction of translation, because molecules are 3-dimensional complexes of atoms that add no thickness to the manifold. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a 3-dimensional smear of atoms no thicker than one atom in its fourth dimension, the direction of its linear translation through 4-space at velocity <math>c</math>. 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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to such 4-dimensional lumps of matter as atomic plasma, and have little experimental knowledge of their internal geometry or processes. We know that such a lump as the sun burns at its surface 3-sphere and emits radiation, and we know a good deal about those surface processes which are nuclear atomic processes, but we know almost nothing about its interior 4-ball, a dimensionally rounder enclosed space whose existence we did not suspect. Every moving surface boundary of matter in the observed universe is evolving in four dimensions at velocity <math>c</math>. The current location in 4-space of any of its material objects corresponds to the present moment in the proper time of that object's inertial reference frame. The object's direction of movement at velocity <math>c</math> corresponds to its proper time dimension, which is a spiral over time, not a Euclidean (straight-line) dimension, since its direction is changing in its orbit. Objects with mass of all sizes, from subatomic particles to the largest objects observed in the cosmos, are perpetually in inertial rotational motion in some orbit, and simultaneously in inertial translational motion propagating themselves through 4-space, in two completely orthogonal inertial motions each at the constant universal rate of transformation <math>c</math>. Every object with mass moves relative to universal 4-coordinate Euclidean space on its own distinct geodesic spiral, a screw displacement trajectory that is the compound of its two completely orthogonal inertial motions, a rotation and a translation. Objects without mass such as photons lie off such moving surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in translational motion at velocity <math>c</math> through all four dimensions concurrently, without any rotational component of motion, so they move through 4-space on straight lines at a compound velocity. The propagation speed of light measured on a straight line through Euclidean 4-space is <math>c\prime = 2c</math>, so we can see in four dimensions, even though we are physically confined to a 3-dimensional manifold that is moving at velocity <math>c</math>. For example, we can look across the center of the mostly-empty 4-ball containing our galaxy and see stars in the opposite sides of its concentric 3-sphere orbital surfaces. We have been unaware that when we look up at night we see stars and galaxies, themselves large 4-dimensional objects, distributed all around us in 4-dimensional Euclidean space, and moving through it, like us, at the constant velocity <math>c</math>. They move in the 4-space direction corresponding to their proper time, perpendicular to all three dimensions of their proper 3-space, and generally the farther they are from us the greater the divergence of their direction of motion from our direction of motion: the greater the relative motion between us and their Hubble redshift. Light from them reaches us directly, propagating on straight lines through 4-space at twice the velocity at which they, and we ourselves, are propagating through 4-space. This physical model of the observed universe is compatible with the theories of special and general relativity, and with the atomic theory of quantum mechanics. It explains those theories geometrically, as expressions of intrinsic symmetries in Euclidean space. == Symmetries == It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway, Burgiel & Goodman-Strauss|2008}} As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s. Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. [[W:Coxeter group|Coxeter's theory of symmetry groups]] generated by reflections did for geometry what Noether's theorem and Einstein's relativity did for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that correspond to distinct symmetry groups, and that their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (their congruent transformations) in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}} <blockquote>Let <math>\mathrm{Q}</math> denote a rotation, <math>\mathrm{R}</math> a reflection, <math>\mathrm{T}</math> a translation, and let <math>\mathrm{Q}^q \mathrm{R}^r\mathrm{T}</math> denote a product of several such transformations, all commutative with one another. Then <math>\mathrm{RT}</math> is a glide-reflection (in two or three dimensions), <math>\mathrm{QR}</math> is a rotary-reflection, <math>\mathrm{QT}</math> is a screw-displacement, and <math>\mathrm{Q^2}</math> is a double rotation (in four dimensions).<br> Every orthogonal transformation is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r</math><br> where <math>(2^q + r \le n)</math>, the number of dimensions.<br> Transformations involving a translation are also expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}</math><br> where <math>(2^q + r + 1 \le n)</math>.<br> For <math>(n = 4)</math> in particular, every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> If we begin with this most elemental [[w:Kinematics|kinematics]] of Coxeter's, and also assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either a <math>\mathrm{Q^2}</math> or a <math>\mathrm{QT}</math>, because we can view any <math>\mathrm{QT}</math> as a <math>\mathrm{Q^2}</math> in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a <math>\mathrm{Q^2}</math>. By the same principle, we can view any <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> as an isoclinic (equi-angled) <math>\mathrm{Q^2}</math> by proper choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} Coxeter's relation is thus a mathematical statement of the principle of relativity, on group-theoretic grounds. It correctly captures the limits to [[W:General relativity|general relativity]], in that we can only exchange the translation (<math>\mathrm{T}</math>) for ''one'' of the two rotations (<math>\mathrm{Q}</math>). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (<math>\mathrm{Q}</math>) up to uncertainty, and can always distinguish the direction of their own proper time translation (<math>\mathrm{T}</math>). As I understand Coxeter theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic methods rather than by Clifford's algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional geometric objects (4-polytopes), and nature can be understood in terms of their [[W:group action|group actions]], including centrally their group <math>SO(4)</math> [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic <math>SO(4)</math> rotational expressions as the [[W:Regular_4-polytope|regular 4-polytopes]]. Their discrete isoclinic rotations are distinguishing properties of fundamental objects in geometry, relativity and quantum mechanics. For example, atoms exhibit <math>SO(4)</math> symmetries of the discrete isoclinic (equi-angled) double rotations (<math>\mathrm{Q^2}</math>) of a set of regular 4-polytopes that is characteristic of their [[w:Atomic_number|atomic number]]. == Special relativity describes Euclidean 4-space == <blockquote>Our entire model of the universe is built on symmetries. Some, like isotropy (the laws are the same in all directions), homogeneity (same in all places), and time invariance (same at all times) seem natural enough. Even relativity, the Lorentz Invariance that allows everyone to observe a constant speed of light, has an elegance to it that makes it seem natural.<ref>{{Cite book|first=Dave|last=Goldberg|title=The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality|chapter=§10. Hidden Symmetries: Why some symmetries but not others?|year=2013|publisher=Dutton Penguin Group|isbn=978-0-525-95366-1|ref={{SfnRef|Goldberg|2013}}}}</ref></blockquote> Although the Minkowski spacetime of relativity is a non-Euclidean 4-dimensional space,{{Efn|Spacetime is a non-Euclidean (curved) 4-dimensional space because it consists of three orthogonal space dimensions and a time dimension. The time dimension is not orthogonal to the three spatial dimensions; the time coordinate has the opposite sign as the three space coordinates so spacetime is hyperbolic, not a Euclidean (flat) 4-space at all.}} it has been noticed that its 3-dimensional space component could be modeled as a [[W:3-sphere|3-sphere]] embedded in 4-dimensional Euclidean (flat) space. That is, we could imagine that the ordinary 3-dimensional space we perceive is the curved 3-dimensional surface of a 4-dimensional ball (since the surface of a 4-ball is a curved 3-dimensional space called a 3-sphere, just as the surface of a 3-ball like the earth is a curved 2-dimensional space called a 2-sphere). This was [[#Origins of the theory|imagined by Einstein]] himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction. Subsequently it was noticed by others (not mainstream physicists) that if physical space were really embedded in Euclidean 4-dimensional space (with our 3-dimensional space embedded in 4-space as some non-Euclidean 3-manifold, not necessarily a 3-sphere), then the Lorentz transformation effects of special relativity (spatial forshortenings and time dilations and so forth) could all be explained by ordinary perspective geometry in 4-dimensional Euclidean space. Special relativity reduces to classical vector space geometry (based on the 4-dimensional version of the Pythagorean theorem), but if and only if every observer is moving through 4-space at a universal constant velocity <math>c</math>, in some 4-space direction.{{Sfn|Yamashita|2023}} This counter-intuitive alternative geometric model of relativity, which has usually been called [[W:Formulations of special relativity#Euclidean relativity|Euclidean relativity]], is motivated by the fact that in every kind of relativity, but originally in Einstein's special relativity, each observer moves on a vector through a Euclidean four-dimensional space consisting of their three proper spatial dimensions and their proper time dimension, and the Pythagorean vector-sum of their motion through this kind of proper 4-space is always <math>c</math>, as measured by all observers from any inertial reference frame.{{Efn|Each observer is stationary in space in their own proper reference frame, while they move at maximum velocity {{Math|c}} through their own proper time. Observed from another reference frame in relative motion, they appear foreshortened in space in their direction of motion, and their clocks appear to be slowed to less than the maximum velocity {{Math|c}}.}} This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source.{{Efn| ...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity {{Math|c}} through space and proper time.}} But no physicists have taken the leap of claiming that therefore, our universe is physically [[W:Euclidean geometry#Higher dimensions|this kind of Euclidean 4-space]], and that observers are actually moving through it at velocity <math>c</math>. In physics as it has been universally understood, observers are not supposed to be able to move at velocity <math>c</math>. Their motion takes place in 3-space and in universal coordinate time (in Minkowski spacetime), and the cosmos is considered to be a non-Euclidean 3-space, generally a closed (finite) expanding 3-space, but with only three spatial dimensions, not four. In the Euclidean relativity alternative view, however, every observer is always moving at velocity <math>c</math> through the universe, which is real Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. The direction in which they are moving is called their proper time axis.{{Efn|Time in spacetime is universal coordinate time, but there is another kind of time in relativity, the proper time in each inertial reference frame. Your proper time is the time you experience, and every observer has his own proper time; proper time runs at different rates in different inertial reference frames. It runs slower (compared to universal coordinate time) in a gravitational field (according to general relativity), and observers in motion with respect to each other view each other's clocks as running slower than their own clocks (according to special relativity).}} Their movement in time is not just modelled as movement in an abstract fourth dimension (as it is in Minkowski spacetime), their movement in time is isomorphic to their movement through physical space in a distinct direction at velocity <math>c</math>. Two observers' directions of movement through space may be different (or not, if they happen to be going in the same direction). Your proper time dimension is whichever direction you are moving. The other three directions perpendicular to your proper time axis are the three dimensions of your ordinary proper space, which again, will be different directions for you than for other observers moving in a different direction. There are four orthogonal spatial dimensions which we all share, but we share the same orthogonal proper time axis and ordinary proper 3-space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity <math>c</math>, in the same inertial reference frame. Your proper 4-space coordinate system is rotated with respect to another observer's proper 4-space coordinate system, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other, but there are no metric distortions (no Lorentz transformations) between your proper Euclidean 4-space coordinate systems; you are both embedded in the same Euclidean 4-dimensional space <math>\mathbb{R}^4</math>. Lorentz transformations are required only to convert between your respective proper Minkowski spacetime coordinate systems. So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity <math>c</math> through Euclidean 4-space, along with all the masses in its vicinity that are going in (nearly) the same direction. The entire solar system, for example, must be translating in the fourth dimension at the "speed of light" <math>c</math>, although we do not notice it, since we are all moving in that same direction together. Acceleration of an object varies its direction of motion through 4-space, but never its velocity, which is invariant for all objects with mass. Two objects which are in motion relative to each other are both actually in motion at the same velocity <math>c</math>, but in at least slightly different directions.{{Efn|The angular divergence between two observer's motion vectors is proportional to their relative velocity: the more they diverge, the greater their relative velocity, up to the maximum divergence possible in the space. In Euclidean relativity all observers are in motion at velocity {{Math|c}} relative to universal Cartesian 4-coordinate space, so the maximum relative velocity between two observers is {{Math|2c}} when they are moving in exactly opposite directions in 4-space. This is not a contradiction of special relativity, which limits the maximum relative velocity between two observers to {{Math|c}}, it is the same measurement in different units. Special relativity measures all velocities in Minkowski spacetime (in a non-Euclidean 3-space). Euclidean relativity measures all velocities in Euclidean 4-space.}} In Einstein's relativity, the invariant <math>c</math> is the speed of light through 3-space. In Euclidean relativity, the invariant <math>c</math> is the speed of matter through 4-space! The speed of light through 3-space is also perceived as <math>c</math> by all observers, because they are each living in a moving 3-manifold that is translating through 4-space at velocity <math>c</math>. Despite their extreme differences in viewpoint, Einstein's relativity and Euclidean relativity are equivalent theories in complete agreement with each other, by definition. The two theories make exactly the same special relativity predictions of how observers in different inertial reference frames will perceive each other's motions in time and space. It is beyond the scope of this present paper to show how they also agree on the predictions of general relativity, but it is important to understand that our formulation of Euclidean relativity requires our acceptance of the experimentally verified findings of special relativity, general relativity and quantum mechanics. Our model and the standard model both describe the same geometric relations of space and time based on the same evidence, but as embedded in two very different universal host spaces: Euclidean 4-space versus Minkowski spacetime. In some instances our model provides a geometric explanation for physical phenomena where the standard model has none, as yet. An example is the two models' differing explanations for the gravitational coherence of galaxies: either their inertial motion at velocity <math>c</math>, or their possession of experimentally undetected dark matter. A more central example is our use of Coxeter's kinematics to explain the laws of relativistic motion geometrically. {{Efn|More generally, our model is concerned with group-theoretic properties of number elucidated by Coxeter and other geometers. ...cite Kappraff & Adamson's 2003 paper on The Relationship of the Cotangent Function to Special Relativity Theory, geometry and properties of number,{{Sfn|Kappraff & Adamson|2003|loc=Special Relativity Theory, Geometry and properties of number}} which shows how the Lorentz coefficient is a function of a deep geometric property of number{{Sfn|Kappraff & Adamson|2000|loc=A Fresh Look at Number}} discovered by Steinbach,{{Sfn|Steinbach|1997|loc=Golden Fields: A Case for the Heptagon}} by means of which the root formula of geometry in any Euclidean dimension, the Pythagorean theorem, may be derived solely in terms of the addition of polygon side lengths, without recourse to their products or squares. More generally, Steinbach found that in the relations among regular polytope chords, to add is to multiply; every chord is both the product (quotient) of a pair of chords and the sum (difference) of another pair of chords.}} Euclidean relativity is not even a fringe theory; no physicists or astronomers have adopted it. There are many good reasons why the revolutionary leap to a four orthogonal spatial dimensions viewpoint has not been taken, beginning with the universally observed fact that we can only construct three perpendiculars through a point in our immediate space, which appears to be resolutely 3-dimensional, not 4-dimensional. Euclidean relativity offers a nice geometric explanation of the reasons for the Lorentz transformations, but only at the cost of raising other mysteries, which have been difficult for its aficionados to explain. Another mystery is how light signals between observers in relative motion could "catch up" with the receiver moving on a diverging path through 4-space from the emitter. If both observers are already moving at <math>c</math> (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than <math>c</math>. Euclidean relativity is a revolutionary theory indeed, in which <math>c</math> cannot possibly be the speed of light! We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for <math>\mathbb{R}^4</math> to be our real space and not merely an abstract mathematical space), the speed of light through Euclidean 4-space must be <math>c\prime = 2c</math>, with massless photons translating through 4-space at twice the speed of mass-carrying objects. Photons must translate the diagonal distance through 4-space along the long diameter of a unit 4-hypercube, in the same time that massive particles translate inertially along the edge (or the radius) of a unit 4-hypercube. This is conceivable in Euclidean 4-space (and in no other Euclidean space of any dimensionality) because the long diagonal of the unit 4-hypercube is the natural number <math>\sqrt{4}</math>. == An object's motion in space is the product of its discrete self-reflections == Coxeter theory describes all the possible motions of an object in space as local functions of the object's discrete geometry (its shape). Coxeter observed that in a Euclidean space of any number of dimensions, any displacement of a geometric object from one place to another, and any rotation of the object from one orientation to another, can be broken down into the product of a number of discrete self-reflections. Any action of a polytope that transforms its position and orientation in space may be measured as a distinct sequence of self-reflections of the object in its own surfaces. Therefore any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete sequence of local self-reflections. This is a purely kinematic description of motion which does not give any account of forces acting on the object that cause it to move. Coxeter found that both changes in position (translations) and changes in orientation (rotations) can be broken down into the simplest of all displacements (self-reflections). A translation occurs when an object self-reflects twice, in two distinct surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which intersect each other. When a object self-reflects once, it turns itself inside out (it reverses its chirality), but in translations and rotations it self-reflects twice, preserving its chirality. Coxeter's laws of kinematics are a geometric counterpart to Newton's algebraic laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as geometric pictures, which may help us to visualize the relationships between translation and rotation in classic Newtonian physics problems, such as the [[W:Kepler problem|Kepler problem]] of the motion of the planets.<ref>{{Cite book|last=Feynman|first=Richard|title=Feynman's Lost Lecture: The Motion of Planets Around the Sun|date=1996|publisher=W. W. Norton & Company|isbn=978-0393039184}}</ref> But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. In particular, they give us geometric pictures of all the possible motions of objects in 4-dimensional Euclidean space: <blockquote>Every orthogonal transformation in 4-space is expressible as:<br> :<math>\mathrm{Q}^q \mathrm{R}^r \mathrm{T}^t</math><br> where <math>(2^q + r + t \le 4)</math>. Every displacement is either a double rotation <math>\mathrm{Q}^2</math>, or a screw-displacement <math>\mathrm{QT}</math> [where the rotation component <math>\mathrm{Q}</math> is a simple rotation, but the <math>\mathrm{QT}</math> is chiral like a <math>\mathrm{Q^2}</math>]. Every enantiomorphous transformation in 4-space (reversing chirality) is a <math>\mathrm{QRT}</math>.</blockquote> While this description should be understood as geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. Rotation (<math>\mathrm{Q}</math>), reflection (<math>\mathrm{R}</math>) and translation (<math>\mathrm{T}</math>) are obvious analogues of what they are in three-dimensional space, but double rotation (<math>\mathrm{Q}^2</math>) is something new and unprecedented in our physical experience, because double rotations cannot occur until there are four or more dimensions of space in which to rotate. A point in 4-space may be in motion on two circles in two completely orthogonal rotation planes simultaneously. Nothing constrains the two circles to be of the same radius, or constrains either circle from being of infinite radius such that the point moves in a straight line within its rotation plane: a translation <math>\mathrm{T}</math> rather than a rotation <math>\mathrm{Q}</math>. The motion of a rigid object defined by multiple points is constrained by its rigidity, such that the radii of both circles may approach infinity, but only one of the two radii may actually reach infinity. For a rigid object, only one of the two rotations may actually be a translation. Four dimensional Euclidean space can accomodate no such thing as a <math>\mathrm{Q}^2\mathrm{T}</math>. A translating double rotation <math>\mathrm{Q}^2</math>, like a translating simple rotation <math>\mathrm{Q}</math>, is simply a screw translation <math>\mathrm{QT}</math>. That is the essence of the way the principle of relativity operates in Euclidean 4-space. An object which is double-rotating in its own proper inertial reference frame may be seen to be performing nothing more mysterious than a screw translation, when observed from an inertial reference frame from which the object is in relative translational motion. Of course, it is a screw translation in 4-space, which is more mysterious than a screw translation in our ordinary 3-space. In 3-space the orbit of a screw translation (or any <math>\mathrm{Q}</math>) lies on a great circle of an ordinary sphere (a 2-sphere). In 4-space the orbit of a screw translation <math>\mathrm{QT}</math> or double rotation <math>\mathrm{Q}^2</math> lies on a great circle of a hypersphere (a 3-sphere). Such 4-dimensional great circles are inherently more complex than the ordinary circles we experience in 3-space. They do not lie in a single plane, but bend through all four dimensions, in either a left or right hand direction of curvature. Unlike ordinary circles they are inherently chiral, and their circumference is not <math>2\pi r</math>, but like ordinary circles they are truly circular, geodesic circles. In 4-space they might even be said to be more circular than ordinary circles, which bend in only two dimensions. We refer to such 4-dimensional great circles as ''isoclines'', to distinguish them from ordinary 2-dimensional great circles in the plane. Their name comes from the fact that a point on a rigid 4-dimensional object undergoing [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] traverses an isocline circle. In a universe which is real Euclidean four-dimensional space <math>\mathbb{R}^4</math>, wherein these motions occur only at the constant velocity <math>c</math>, there is a universal Cartesian 4-coordinate reference frame in which every rigid object 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, but it is the common reference frame of all observers. == Light propagates through 4-space at twice its apparent velocity <math>c</math>== Coxeter's laws of kinematics in 4-dimensional Euclidean space apply to all objects with mass, but we find there is an additional kind of displacement which applies only to massless particles such as photons, considered as single point objects. Light quanta (photons) translate through 4-space by 4-dimensional reflection <math>\mathrm{R}^4</math>, which may be termed a double translation <math>\mathrm{T}^2</math>, a pure translation via two pairs of parallel reflections without any rotation component <math>\mathrm{Q}</math>. Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by <math>\mathrm{QT}</math>, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic <math>\mathrm{Q^2}</math>, an isoclinically rotating object such as an atom. A simple rotation <math>\mathrm{Q}</math> or simple translation <math>\mathrm{T}</math> is a double reflection <math>\mathrm{R^2}</math>, so a <math>\mathrm{QT}</math> or <math>\mathrm{Q^2}</math> is an <math>\mathrm{R^4}</math>, but not with the same group of reflection angles as a light signal <math>\mathrm{R^4}</math>. A translation <math>\mathrm{T = R^2}</math> is a double reflection in two parallel planes, and a rotation <math>\mathrm{Q = R^2}</math> is a double reflection in two intersecting planes, as in a <math>\mathrm{QT = R^4}</math> which is both at once. A double translation <math>\mathrm{T^2 = R^4}</math> is two double reflections in pairs of parallel planes at once, a reflection in four non-intersecting parallel planes; it is all translation and no rotation. In a <math>\mathrm{T^2}</math> all the motion goes to translation, so the translation goes twice as far as the simple translation <math>\mathrm{T}</math> in a <math>\mathrm{QT}</math>. A double translation <math>\mathrm{T^2 = R^4}</math> is the opposite of a double rotation <math>\mathrm{Q^2 = R^4}</math>, which is stationary but rotates twice as far as the simple rotation <math>\mathrm{Q}</math> in a <math>\mathrm{QT}</math>. The product of the two translations in a <math>\mathrm{T^2}</math> is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple <math>\mathrm{T}</math> over the edge length (or radius) of the unit 4-hypercube. The [[w:Tesseract|4-hypercube (also known as the 8-cell or tesseract)]] is ''radially equilateral'', which means its edge length is equal to its radius, like the hexagon, so its long diameter (twice its radius) is exactly twice its edge length. The photon moves an equal distance in four orthogonal directions. By the four-dimensional Pythagorean theorem, each of those four distances is half the total distance the photon moves: one edge length (one radius) is half the total diagonal distance moved (the long diameter). That total movement is a double the unit-distance translation, but without any rotation component, so it cannot carry any mass with it. A <math>\mathrm{T^2}</math> cannot reposition a 4-polytope the way a <math>\mathrm{QT}</math> does, it can only reposition a quantum of energy that has no distinguishing rotational symmetry, such as a photon. That is the price light pays to move exactly twice as fast as matter.{{Efn| ...lensing of double translations <math>\mathrm{T^2 = R^4}</math> in more than two pairs of parallel planes at once...relationship to the frequency of light emitted and the coherence length of the wave packet...}} == Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions == {{Efn|...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...}} Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space Cartesian coordinates of objects in the scene being observed. But if all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) in correct scene construction, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a unified spacetime to explain 4-dimensional perspective effects at high relative velocities, because Euclidean 4-space is already 4-dimensional, and those effects fall out naturally from the 4-dimensional Pythagorean theorem, exactly as ordinary visual perspective does in three dimensions from the 3-dimensional Pythagorean theorem. Because one of the four spatial dimensions corresponds to an observer's direction of motion (in both space and proper time), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in a spatial dimension. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity the actual rate of physical processes varies from place to place, and reference frame differences are neither reciprocal nor illusory.) None of these Lorentz effects are beyond geometric explanation or paradoxical. The universe is unexpectedly strange to us in precisely the ways the Euclidean fourth dimension is strange to us; but that does hold many surprises. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way 3-space is much more interesting and deeply explanatory to us than it would be if we experienced it only as a 2-space with many folds and curves, as perhaps an ant does. The emergent properties of 4-space are hard for us to visualize because they lie so wholly beyond our physical experience, just as it was hard for our ancestors to imagine the earth as round like a ball. However, Euclidean spaces of successive dimensions are analogous, and so higher dimensional spaces can be anticipated and explored: that is Schläfli's great discovery. Moreover dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic group-theoretic symmetries: that is Nother's great discovery. == Dimensional relativity == Coxeter's kinetic law of <math>n</math>-dimensional congruent Euclidean transformations 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated. By analogy to the surface of the earth, the bounding surface of a spherical region of <math>n</math>-dimensional Euclidean space is an <math>(n-1)</math>-sphere, a spherical space of one fewer dimensions than the <math>n</math>-ball of Euclidean space it surrounds. In dimensional relativity the sky is not a ceiling, but an infinite regress of alternating spherical and Euclidean <math>n</math>-spaces of increasing <math>n</math>, accessible from each observer's point of view. By dimensional analogy, each observer looks up into their own reference frame's regress of concentric alternating <math>n</math>-spaces. By the capacity for dimensional analogy which they possess, some observers see deeper into <math>n</math>-dimensional space than others. == Polycentric spherical relativity == An intelligent observer equipped with the principle of relativity may perceive the universe from any inertial reference frame, not only from their own proper stationary reference frame in the <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+1)</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. It is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in non-Euclidean spacetime, and Coxeter's dimensional relativity of orthogonal group actions in Euclidean and spherical spaces of any number of dimensions. It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: "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.]}} == Revolutions == The original Copernican revolution in 1543 displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the earth performing a ''revolution'' around the sun, and the stars remaining on a fixed 2-sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. In such fashion the Euclidean four-dimensional revolution, emerging three to five centuries later, initially lends itself to the big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the galaxies need not be equidistant from a single origin in time, any more than all the stars lie in the same galaxy, equidistant from a single center in space. The expanding sphere of matter on the surface of which we find ourselves living is likely to be one of many 3-spheres expanding at velocity <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 we 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 this work we suggest a theory of ''n''-dimensional real space and how light travels in it, a theory which says we can see into four orthogonal dimensions of Euclidean space, and so when we look up at night we see cosmological objects distributed in at least four dimensions of space around us, rather than all located in our own local 3-space. Looking still deeper and farther out, the universe viewed as a 4-sphere might, or might not, be expanding, and the most distant objects we see when we look up at night may, or may not, lie in our 4-dimensional hyperplane. Real space has ''n'' dimensions as [[w:Hermann_Grassmann|Grassmann]] and [[w:Schläfli|Schläfli]] showed, and we do not know how many dimensions the most distant objects we see may be distributed in. They need not all lie within the four spatial dimensions in which we have now learned to observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system. When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us 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 ''n''-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 much 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 translations), nearby objects must be translating radially at the same velocity, since the objects in a system (such as our solar system or galaxy) do not separate rapidly over time but remain in orbital formation. Each system's screw 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 each galaxy's own distinct 3-sphere is expanding at this great rate from its galactic center. The standard theory has been that the entire observable universe is expanding from a single big bang origin in time, with galaxies forming later. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also supports theories which require no single origin point in space and time. These are the voyages of starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed long ago, in whatever big bang our atoms emerged from, and have never slowed down since. == Origins of the theory == Einstein himself may have been the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean 3-sphere, in what was narrowly the first written articulation of the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space as embedded in Euclidean 4-space.}} Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.<ref>{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}</ref> He invited us to imagine "A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions", but he was careful to disclaim parenthetically that "The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice." Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal to that disclaimer of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from their perspective; the foreshortenings, clock desynchronizations and other Lorentz transformations it predicts are proper calculations of actual perspective effects; but real space is a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four spatial dimensions.'' The Euclidean theory of relativity differs from the special theory of relativity in ascribing to the physical universe a geometry of four or more orthogonal spatial dimensions, rather than the special theory's [[w:Minkowski spacetime|Minkowski spacetime]] geometry, in which three spatial dimensions and a time dimension comprise a unified 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 the 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 which is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the direct 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 interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schläfli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it that way, is that there ''is'' a boundary between three and four dimensional space. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our physical world apparently only three dimensional? Why would it have just ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schläfli mapped? What is the nature of the boundary which confines us to just three dimensions? We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary surface. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way, by receiving light signals that travelled through it to us on straight lines. In that case the reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed all around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creatures, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not perplex us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell. Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schläfli discovered something else: all the astonishing regular objects that exist in higher dimensions, which vastly extend our notions of the beauty and mystery of space itself, and the intrinsic spatial symmetries of our universe which geometry reveals. Space is more commodious than we thought it was, and permits previously unimagined 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 an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation? We must look for a physically powerful explanation in the geometry of space itself, which general relativity properly associates with the gravitational or inertial force. <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. 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. 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}} k7yen60dkl33qjth4y842e2at8mei22 0 296128 2835586 2534105 2026-10-02T22:59:34Z Scogdill 1331941 2835586 wikitext text/x-wiki ==Also Known As== * Lily Le Breton * Lillie Le Breton * Lily Langtry * Emilie Charlotte, Lady de Bathe * The "Jersey Lily" * Family name: Le Breton, Langtry ==Overview== ==Acquaintances, Friends and Enemies== ===Acquaintances=== * Frank Miles * Thomas Heron Jones, 7th Viscount Ranelagh * Rupert Potter * Edward Poynter * Edward Burne-Jones ===Friends=== * Oscar Wilde<ref name=":0" /> * John Everett and Effie Millais<ref name=":1">{{Cite journal|date=2023-03-13|title=Edward Langtry|url=https://en.wikipedia.org/w/index.php?title=Edward_Langtry&oldid=1144469970|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_Langtry.</ref> * Henrietta Labouchère (and then after May 1883 enemy)<ref name=":0" /> * William Gladstone (1809–1898)<ref name=":0" /> === Sexual Partners === * [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]]<ref name=":0" /> * Lord Shrewsbury<ref name=":0" /> * Prince Louis of Battenberg<ref name=":0" /> * Frederick Gebhard * ===Enemies=== ==Organizations== ==Timeline== '''1874 March 9''', the day of Lily le Breton Langtry's wedding. In her memoirs, Lady Lucy ("Lucille") Gordon Duff writes about Lily Langtry. Her family had just moved to Jersey. Lucy Christiana Wallace was 11 years old,<ref>"Lucy Christiana Wallace Sutherland." {{Cite web|url=https://www.thepeerage.com/p18117.htm#i181170|title=Person Page|website=www.thepeerage.com|access-date=2023-07-01}} https://www.thepeerage.com/p18117.htm#i181170.</ref> and Lily le Breton had just married:<blockquote>When we arrived the harbour was decorated with flags and garlands of flowers were hanging from all the windows. We were told it was to celebrate the wedding of the Dean's daughter, Lily le Breton, the most beautiful girl in Jersey, who had been married that morning to Mr. Langtry and had gone away to London “to be a great lady”. We were both very curious about this lovely creature, stories of whose romantic career as a court beauty used to be circulated all over the Island. We were always hearing of what she had worn at the opera, of how she had set a new fashion in hats, and of how often the Prince of Wales had danced with her at the Devonshire House ball. When she used to come back to stay at hex old home we were all agog with [24/25] excitement, and receptions were given in her honour at Government House* Nearly half the population used to go down to the Quay to cheer the boat which brought her ashore, and flags would be flying all along her route. It was almost like the arrival of a queen. Elinor and I determined that come what might we would see the "Jersey Lily” and find out whether she was as beautiful as we had been told. It required some ingenuity on our part, as although we were very friendly with the Governor's family and his daughter, Ada, was our special playmate, we were too young to attend any of the formal parties at which Mrs. Langtry would be a guest. In the end we evolved the plan of hiding in the rooms in which she would take off her cloak at one of the receptions. We stole out of our beds and dressing in the dark for fear of arousing suspicion if a light were noticed in our room, made our way to Government House. Ada let us in, and we crept upstairs like conspirators and hid under the dressing-table, draping the folds of white muslin, with which it was trimmed, so that they completely screened us, leaving- only a tiny peephole. We had some time to wait, for Lily Langtry always made her entry last of all the guests at a party, and one by one the women came upstairs to remove cloaks and shawls. It was the first time we had seen most of them in evening dress, and we had a great deal of fun comparing their dresses. Just as we were beginning to feel intolerably cramped from our uncomfortable position crouched on the floor, a buzz of conversation and a light and very musical laugh on the stairs announced the arrival of Lily Langtry, and a moment later she came into the room. I never saw any woman more divinely lovely than [25/26] she looked in her white dress, with a scarlet flower in her hair nestling against one ear. She came and sat down at the dressing-table while she arranged her dress and pinned a beautiful diamond brooch on one shoulder. We were so close to her that we dared not move one inch, for fear of touching her dress and giving away the secret of our presence, but even through the folds of the muslin curtain we could see her perfect beauty. There was an extraordinary radiance about Lily Langtry that I have never seen in any other woman, and there was something so vital and magnetic in her personality that a room seemed empty when she left it.<ref>Gordon Duff, Lucy (née Wallace) Sutherland "Lucille." ''Discretions and Indiscretions''. Jarrolds, 1932. Internet Archive https://archive.org/embed/in.ernet.dli.2015.208501 (Retrieved July 2023).</ref> (24–26)</blockquote>'''1876''', Lily Langtry and her husband moved to London.<ref name=":0">{{Cite journal|date=2023-06-25|title=Lillie Langtry|url=https://en.wikipedia.org/w/index.php?title=Lillie_Langtry&oldid=1161904034|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lillie_Langtry.</ref> '''1877, late, to June 1880''', Lily Langtry and the Prince of Wales had a sexual relationship. '''1879 July''', Lily Langtry and [[Social Victorians/People/Shrewsbury|Charles Chetwynd-Talbot, Lord Shrewsbury]] began a sexual relationship; he was 18 and she was 25.<ref name=":2">{{Cite journal|date=2023-06-10|title=Charles Chetwynd-Talbot, 20th Earl of Shrewsbury|url=https://en.wikipedia.org/w/index.php?title=Charles_Chetwynd-Talbot,_20th_Earl_of_Shrewsbury&oldid=1159508278|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Charles_Chetwynd-Talbot,_20th_Earl_of_Shrewsbury.</ref> '''1880 January''', Lily Langtry and [[Social Victorians/People/Shrewsbury|Charles Chetwynd-Talbot, Lord Shrewsbury]] planned to run away together, but he began a relationship with Ellen Palmer-Morewood,<ref name=":2" /> who was married and whom he married 21 June 1882, and whose son Charles John Alton Chetwynd-Talbot was born 2 months later, on 8 September 1882.<ref>"Charles Henry John Chetwynd-Talbot, 20th Earl of Shrewsbury." {{Cite web|url=https://www.thepeerage.com/p1233.htm#i12324|title=Person Page|website=www.thepeerage.com|access-date=2023-07-01}} https://www.thepeerage.com/p1233.htm#i12324.</ref> '''1881 March 8''', Lily Langtry was in Paris with Arthur Jones for the birth of Jeanne Marie Langtry; the pregnancy had ended the relationship with the Prince of Wales.<ref name=":1" /> Jones was probably Jeanne Marie's father.<ref name=":0" /> '''1882''', Pears Soaps used Lily Langtry's face in its advertising, making her "the first celebrity to endorse a commercial product."<ref name=":0" /> ==== 1890 January 28, Tuesday ==== Lily Langtry was too ill to act in the opening performance at the St. James's Theatre:<blockquote>Mrs. Langtry’s inability to appear at the St. James on Tuesday evening disappointed a distinguished company of "first-nighters." Not only had the Prince of Wales undertaken to be present, but among those who had arranged to "assist" on the occasion were Lord and Lady Coleridge, Lady de Grey, [[Social Victorians/People/Mandeville|Lady Mandeville]], Baroness Stern, Lord Wolverton, Sir Charles Russell, Mr. Alfred de Rothschild, Mr. and Mrs. Hwfa Williams, Mr. Akers Douglas, Mr. Lockwood, Captain Shaw, Mr. Whistler, Mr. Hamilton Aidé, Mr. and Mrs. George Lewis, Mr. and Mrs. John Hare, &c., &c. As half-past six on Tuesday evening, when Mrs. Langtry’s brother called at her house to take her to the St. James' he found her in a dead faint. When she was brought round she would fain have gone down to the theatre, and was with difficulty dissuaded from so reckless an act. How physically unfit she was to play that evening has since been only too clearly demonstrated. The company were all at the St. James', waiting to go on, and their consternation at the news may be imagined. A recent bulletin in regard to Mrs. Langtry is as follows:— "Acute attack of blood-poisoning, with severe fever; and the doctors compel her to keep her bed for some time." Playgoers will regret to know that the popular actress' condition is so serious. The Prince of Wales has written a sympathetic letter to Mrs. Langtry’s sister-in-law, and sent Sir Oscar Clayton to offer bis professional assistance. Mrs. Langtry, unlike most contemporary actresses has, considering her age, been a comparatively short time in the profession, having made her début at the Haymarket Theatre in December, 1881, when she was almost 30 years old. Miss Ellen Terry, for example, has been on the stage 33 years. Mrs. Kendal, though only four years older than Mrs. Langtry, became an actress almost as soon as she was in short clothes. Patti has been 27 years on the operatic stage, while Mrs. Sterling (now retired) was for nearly 50 years before the public. Miss Nellie Farren, who foots it night after night so nimbly at the Gaiety, was a veteran before Mrs. Langtry was heard of, Mrs. John Wood is no longer in her teens as an actress, and even the younger generation, Miss Amy Roselle, Miss Alma Murray, Miss Fanny Brough, Miss Florence St. John, and a score of others are all Mrs. Langtry’s seniors in point of experience. But few, if any, have accomplished such a feat as the Lily of Jersey, who without serving an apprenticeship, entered the profession at 30, and in half a dozen years realised a fortune.<ref>"Green Room Gossip." ''Sheffield Evening Telegraph'' 1 February 1890, Saturday: 2 [of 4], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000275%2F18900201&page=2&article=003&stringtohighlight=mandeville+viscount. ''The Evening Telegraph and Star'', p. 2. </ref></blockquote>'''1897 May 13''', Lily Langtry successfully obtained an American divorce from Edward Langtry by default because he failed to respond to the summons.<ref name=":1" /> '''1899 July 27''', Lily Langtry and Hugo Gerald de Bathe married. '''1907''', Hugo Gerald de Bathe's father died and he became Lord de Bathe. ==Demographics== * Nationality: British, Island of Jersey ===Residences=== ==Family== *Lily Langry (13 October 1853 – 12 February 1929)<ref name=":0" /> *Edward Langtry (14 February 1846 – 15 October 1897)<ref name=":1" /> *Arthur Clarence Jones (1854–1930), unmarried relationship<ref name=":0" /> *#Jeanne Marie Langtry (8 March 1881 – )<ref name=":0" /> *Hugo Gerald de Bathe (1871–1940) ===Relations=== ==Notes and Questions== == Archives and Memoirs == * Langtry, Lily. ''The Days I Knew''. * Langtry, Emilie Charlotte. ''The life of Mrs. Langtry, the Jersey Lily, and queen of the stage''. Pinder & Howes Leeds, 1882. ==Footnotes== {{reflist}} sc3y49doe0cx8zml7rwstd6x2cei4ri 0 301630 2835503 2585757 2026-10-02T12:41:56Z MrKeefeJohn 2950253 ([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Con Wes Anderson e Noah Baumbach (cropped)2.jpg]] → [[File:Wes Anderson e Noah Baumbact.jpg]] 2835503 wikitext text/x-wiki [[Category:Film]] [[Category:Cinema Aesthetics]] == Overview == [[File:Noah Baumbach Cannes 2017.jpg|Noah_Baumbach_Cannes_2017|thumb|right|thumb|Noah Baumbach]] Noah Baumbach is an American filmmaker known best for writing and directing witty intellectual comedies that are typically set in New York. Growing up, he was inspired by Woody Allen's films. Additionally, he notes Steve Martin as a comedic inspiration.<ref>https://www.indiewire.com/features/general/noah-baumbach-influences-filmmaker-spielberg-scorsese-max-ophuls-1201883274/</ref> == Filmography == ==== Baumbach's Direction ==== Noah Baumbach has directed 12 feature narrative films which are: * ''Kicking and Screaming'' (1995) * ''Mr. Jealousy'' (1997) * ''Highball'' (2002) <small>*film was never fully finished</small> * ''The Squid and the Whale'' (2005) * ''Margot and the Wedding'' (2007) * ''Greenberg'' (2010) * ''Frances Ha'' (2012) * ''While We’re Young'' (2014) * ''Mistress America'' (2015) * ''The Meyerowitz Stories'' (2017) * ''Marriage Story'' (2019) * ''White Noise'' (2022) ==== Baumbach's Writing ==== [[File:Wes Anderson e Noah Baumbact.jpg|Con_Wes_Anderson_e_Noah_Baumbach_(cropped)2|thumb|right|Noah Baumbach and Wes Anderson]] Noah Baumbach writes all of his own films. Noah Baumbach collaborates frequently with [[Wes Anderson]]. He was a writer on the following Anderson's films: ''Fantastic Mr. Fox'' (2009) and ''The Life Aquatic with Steve Zissou'' (2004). <ref>https://www.imdb.com/name/nm0000876/fullcredits</ref> He also works with his partner Greta Gerwig. He was a writer and producer on Gerwig's 2023 smash hit ''Barbie'' (2023). == Filmmaker's Aesthetics == === Mumblecore === Noah Baumbach was an important icon in popularizing the sub-genre of Mumblecore.<ref>https://faroutmagazine.co.uk/noah-baumbach-and-the-importance-of-the-mumblecore-movement/</ref> Mumblecore was inspired by films by Andrei Tarkovsky and the film ''Slacker'' (Linklater 1990). Common aesthetics of mumblecore include natural feeling and sounding dialogue, static camera movements, and natural lighting. ''Frances Ha'', directed by Baumbach and written by himself and Greta Gerwig was one of the first mumblecore films to skyrocket the genre. Even in his more recent films like ''Marriage Story'' (2019) and ''White Noise'' (2022) there are still mumblecore-like aesthetics like lower budget filmmaking techniques about characters struggling through life and leaving their youth. === Common Themes === Baumbach has been called an auteur. He has many reoccuring themes and motifs in his films. ==== Locations Set in NYC ==== Most, if not all of his films are set on the East Coast, specifically New York City. <ref>https://www.youtube.com/watch?v=2Qip9FvdOBM&feature=youtu.be</ref> Thus, he has many scenes set on a train or subways. ==== Divorce and Step Families ==== He is inspired by his own personal experiences and uses it for his films. All of Baumbach's films contain broken families or step families. ==== Prestigious References ==== Noah Baumbach is known for having characters with artistic careers, and these characters having intelligent conversations about high art.<ref>https://www.tandfonline.com/doi/full/10.1080/0950236X.2021.1965293</ref> ==== Moments in His Films ==== Drawing from his love of Steve Martin, Noah Baumbach is known for having moments of screwball comedy in his films; as well as, moments of deep wisdom between the comedy. == Reception/Awards == Noah Baumbach was been nominated for multiple prestigious awards during his career. Some of the highlights include: * 2020 Nominee Oscar - Best Motion Picture of the Year - ''Marriage Story''<ref>https://www.imdb.com/name/nm0000876/awards/</ref> * 2020 Nominee Oscar - Best Original Screenplay - ''Marriage Story'' * 2020 Nominee BAFTA Film Award - Best Screenplay (Original) - ''Marriage Story'' * 2017 Nominee Palme d'Or - ''The Meyerowitz Stories'' * 2006 Nominee Oscar - Best Writing, Original Screenplay - ''The Squid and the Whale'' == References == <references /> [[Category:Film]] [[Category:Cinema Aesthetics]] ienx47kgnwv8y5objwtadmzuzi4s80b 2 303895 2835665 2749214 2026-10-03T10:22:58Z B.NIROSHAN 2980192 2835665 wikitext text/x-wiki '''Hydraulic Energy''' is a form of energy.<ref name=":0">{{Cite book |title=Fluid mechanics |date=2007 |publisher=Pearson/Prentice Hall |isbn=978-0-13-129293-2 |editor-last=Douglas |editor-first=John F. |edition=5. ed., [reprint.] |location=Harlow |page=767}}</ref>The energy that a [[fluid]] may possess is known as hydraulic energy. <ref name=":0" />It might exist as [[Thermal energy|thermal]], [[Kinetic energy|kinetic]], [[Potential energy|potential]], [[Strain energy|strain]], or [[Pressure energy|pressure]]<nowiki/>s energy.<ref name=":0" /> ==See also== * [[hydropower]] ==References== <!-- [[Category:Energy]] --> 0zfq1wsr6yfwdure5wt7q6p1l3fhfno 2835666 2835665 2026-10-03T10:23:20Z B.NIROSHAN 2980192 2835666 wikitext text/x-wiki '''Hydraulic Energy''' is a form of energy.<ref name=":0">{{Cite book |title=Fluid mechanics |date=2007 |publisher=Pearson/Prentice Hall |isbn=978-0-13-129293-2 |editor-last=Douglas |editor-first=John F. |edition=5. ed., [reprint.] |location=Harlow |page=767}}</ref>The energy that a [[fluid]] may possess is known as hydraulic energy. <ref name=":0" />It might exist as [[Thermal energy|thermal]], [[Kinetic energy|kinetic]], [[Potential energy|potential]], [[Strain energy|strain]], or [[Pressure energy|pressure]] energy.<ref name=":0" /> ==See also== * [[hydropower]] ==References== <!-- [[Category:Energy]] --> 70gul05jafze1inh1xspd6q8bgige0v 2 303926 2835663 2610246 2026-10-03T10:21:16Z B.NIROSHAN 2980192 2835663 wikitext text/x-wiki * [[What was the first spear?|What was the first spear? - Wikiversity]] ==Sub Pages== {{Special:Prefixindex/{{FULLPAGENAME}}/|hideredirects=0|stripprefix=1}} 99pkcpcs7ae0c44p6ougmngn1nmdkjf 2835664 2835663 2026-10-03T10:21:58Z B.NIROSHAN 2980192 2835664 wikitext text/x-wiki ==Sub Pages== {{Special:Prefixindex/{{FULLPAGENAME}}/|hideredirects=0|stripprefix=1}} qsy9tby2t3pwfgn6o67hc4rav17a6ej 0 305491 2835645 2821561 2026-10-03T08:17:39Z ~2026-53155-18 3112031 /* */ 2835645 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby Boom |1946-1964 (19) |1946-1964 (19) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1965-1984 (20) |- |Millennial |1980-1994 (15) |1981-1996 (16) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1985-2005 (21) |- |Generation Z |1995-2009 (15) |1997-2012 (16) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2004 (8) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2013-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1964 | rowspan="5" |Generation X !Baby Boom |Generation X |- |1965-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="3" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980 | rowspan="4" |Generation Y !Generation X |Millennial |- |1981 | rowspan="4" |Millennial |Generation X !Millennial |- |1982-1986 | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2012 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2013-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf i8yd37neu8r4ygp0uqoccmad4r0rypm 2835646 2835645 2026-10-03T08:19:01Z ~2026-53155-18 3112031 /* */ 2835646 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby Boom |1946-1964 (19) |1946-1964 (19) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1965-1984 (20) |- |Millennial |1980-1994 (15) |1981-1996 (16) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1985-2005 (21) |- |Generation Z |1995-2009 (15) |1997-2012 (16) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2013-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1964 | rowspan="5" |Generation X !Baby Boom |Generation X |- |1965-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="3" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980 | rowspan="4" |Generation Y !Generation X |Millennial |- |1981 | rowspan="4" |Millennial |Generation X !Millennial |- |1982-1986 | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2012 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2013-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf k30nr4f18v1hw15vljnus41cwcbsgpk 2835647 2835646 2026-10-03T08:25:23Z ~2026-53155-18 3112031 /* */ 2835647 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby Boom |1946-1964 (19) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1966-1985 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1986-2005 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2012 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2013-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf da13wyz9594omks5v01dopayuweqegb 2835649 2835647 2026-10-03T08:27:28Z ~2026-53155-18 3112031 /* */ 2835649 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1945 (22) |- |Baby Boom |1946-1964 (19) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1946-1965 (20) |- |Generation X |1965-1979 (15) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1966-1985 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1986-2005 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2006-mid 2020s (19-21*) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2011 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2012-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf azsv25xy2n0b3sw3b46qz193tu3nevv 2835650 2835649 2026-10-03T08:30:11Z ~2026-53155-18 3112031 /* */ 2835650 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1943 (20) |- |Baby Boom |1946-1964 (19) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1944-1963 (20) |- |Generation X |1965-1979 (15) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1964-1983 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1984-2003 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2004-2023 (20) |- |Generation Beta |2025-2039 (15) |TBD |2030-early 2050s (20-25*) |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2011 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2012-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf 6fioue3y6iw3517y6kkodbhyiivxdh4 2835652 2835650 2026-10-03T08:32:33Z ~2026-53155-18 3112031 /* */ 2835652 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1925-1945 (21) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1943 (20) |- |Baby Boom |1946-1964 (19) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1944-1963 (20) |- |Generation X |1965-1979 (15) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1964-1983 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1984-2003 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2004-2023 (20) |- |Generation Beta |2025-2039 (15) |TBD |2030-? |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2011 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2012-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1928-1942 (15) |1925-1945 (21) |1925-1945 (21) |- |Baby Boom |1946-1960 (15) |1946-1964 (19) |1943-1964 (22) |- |Generation X |1965-1979 (15) |1965-1980 (16) |1961-1981 (21) |- |Millennial |1982-1994 (13) |1981-1996 (16) |1980-2005 (26) |- |Generation Z |1995-2012 (18) |1997-2009 (13) |1995-2012 (18) |- |Generation Alpha |2013-2024 (12) |2010-2024 (15) |2010-2029 (19) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf aditqjjm7tah42wvv7hxmogvnc0unoi 2835655 2835652 2026-10-03T08:45:23Z ~2026-53155-18 3112031 /* */ 2835655 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1928-1945 (18) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1943 (20) |- |Baby Boom |1946-1963 (18) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1944-1963 (20) |- |Generation X |1964-1979 (16) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1964-1983 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1984-2003 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2004-2023 (20) |- |Generation Beta |2025-2039 (15) |TBD |2030-? |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2011 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2012-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1925-1945 (21) |1925-1945 (21) |1928-1945 (18) |- |Baby Boom |1946-1963 (18) |1946-1963 (18) |1946-1963 (18) |- |Generation X |1964-1979 (16) |1964-1979 (16) |1964-1981 (18) |- |Millennial |1980-1994 (15) |1980-1995 (16) |1982-1996 (15) |- |Generation Z |1995-2009 (15) |1996-2011 (16) |1997-2011 (15) |- |Generation Alpha |2010-2024 (15) |2012-2024 (13) |2012-2024 (13) |- |Generation Beta |2025-2039 (14) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf q2pr2m1iosc7xy7n8imj32ezj24n1ve 2835656 2835655 2026-10-03T08:47:45Z ~2026-53155-18 3112031 /* */ 2835656 wikitext text/x-wiki {{Dynamic List}} {| class="wikitable" !Generation !Mccrindle !Pew !Strauss-Howe !Japanese Traditional Generations !GenWHY |- |Arthurian | rowspan="19" |N/A | rowspan="19" |N/A |1433-1460 (28) | | |- |Humanist |1461-1482 (22) | | |- |Reformation |1483-1511 (29) | | |- |Reprisal |1512-1540 (29) | | |- |Elizabethan |1541-1565 (25) | | |- |Parliamentary |1566-1587 (22) | | |- |Puritan |1588-1617 (30) | | |- |Cavalier |1618-1647 (30) | | |- |Glorious |1648-1673 (26) | | |- |Enlightenment |1674-1700 (27) | | |- |Awakening |1701-1723 (23) | | |- |Liberty |1724-1741 (18) | | |- |Republican |1742-1766 (25) | | |- |Compromise |1767-1791 (25) | | |- |Transcendental |1792-1821 (30) | | |- |Gilded |1822-1842 (21) | | |- |Progressive |1843-1859 (17) | | |- |Missionary |1860-1882 (23) |明治一桁世代 (Meiji Hitoketa Sedai) 1868-1877 (10) | |- |Lost |1883-1900 (18) |明治二桁世代 (Meiji Futaketa Sedai) 1878-1911 (34) | |- |G.I. |?-1924 |?-1927 |1901-1924 (24) |大正世代 (Taisho Sedai) 1912-1925 (13) | |- |Silent |1928-1945 (18) |1928-1945 (18) |1925-1942 (18) |昭和一桁世代 (Showa Hitoketa Sedai) 1926-1935 (10) 焼け跡世代 (Yakeato Sedai) 1935-1946 (12) |1924-1943 (20) |- |Baby Boom |1946-1963 (18) |1946-1963 (18) |1943-1960 (18) |団塊世代 (Dankai "Boomer" Sedai) 1947-1950 (4) - cusp しらけ世代 (Shirake Sedai) 1951-1964 (14) |1944-1963 (20) |- |Generation X |1964-1979 (16) |1964-1981 (18) |1961-1981 (21) |バブル世代 (Baburu "Bubble" Sedai) 1965-1970 (6) - cusp 氷河期世代** (Hyougakki "Ice Age" Sedai) 1971-1981 (11) |1964-1983 (20) |- |Millennial |1980-1994 (15) |1982-1996 (15) |1982-2005 (24) |プレッシャー世代 (Puresshaa "Pressure" Sedai) 1982-1986 (5) - cusp ゆとり世代 (Yutori Sedai) 1987-1996 (10) |1984-2003 (20) |- |Generation Z |1995-2009 (15) |1997-2011 (15) |N/A |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) 1997-2005 (9) - cusp コロナ世代*** (Korona "COVID" Sedai) 2001-2014 (14) |N/A |- |Generation Alpha |2010-2024 (15) |2012-? |2006-2029 (24) |TBD |2004-2023 (20) |- |Generation Beta |2025-2039 (15) |TBD |2030-? |TBD |TBD |} <nowiki>*</nowiki> = Based on the definition of S&H generations being approximately 20-25 years. <nowiki>**</nowiki> = This can further be divided up into 団塊ジュニア世代 (Dankai Jr. "Boomer Jr." Sedai) (early 1970s) and the rest ポスト団塊ジュニア世代 (Post Dankai Jr. Sedai) (late 1970s-1981) <nowiki>***</nowiki> = (As of 2024,) it is generally accepted as someone who was 5 (turning 6) to 19 years old in 2020 (start of COVID-19 pandemic), which likely affected their school years (K-12 and university). According to a 2022 survey from Japanese high school students, they considered themselves as コロナ世代. (However, it won't be confirmed until c. 2032 when all of them are 18 or older to see if this name stays). In the context of working, however, it can be defined as 1998-2006 (age 18-26 in 2024). This is because during the COVID pandemic (2020-2024), they became adults and were in university during part or all of this time period. There is also a chance that this could be the last Japanese traditional generation name because Japanese people are beginning to use the English generation names (e.g., Z世代・α世代). <nowiki>****</nowiki> = Based on GenWHY generations ranging approximately 20-22 years. {| class="wikitable" ! rowspan="2" |Birth year range ! colspan="5" |English Generation !Japanese Generation |- !McCrindle !Pew !Strauss-Howe ! colspan="2" |Summary ! |- |1433-1460 | rowspan="23" |N/A | rowspan="25" |N/A |Arthurian | colspan="2" |Arthurian | rowspan="18" |N/A |- |1461-1482 |Humanist | colspan="2" |Humanist |- |1483-1511 |Reformation | colspan="2" |Reformation |- |1512-1540 |Reprisal | colspan="2" |Reprisal |- |1541-1565 |Elizabethan | colspan="2" |Elizabethan |- |1566-1587 |Parliamentary | colspan="2" |Parliamentary |- |1588-1617 |Puritan | colspan="2" |Puritan |- |1618-1647 |Cavalier | colspan="2" |Cavalier |- |1648-1673 |Glorious | colspan="2" |Glorious |- |1674-1700 |Enlightenment | colspan="2" |Enlightenment |- |1701-1723 |Awakening | colspan="2" |Awakening |- |1724-1741 |Liberty | colspan="2" |Liberty |- |1742-1766 |Republican | colspan="2" |Republican |- |1767-1791 |Compromise | colspan="2" |Compromise |- |1792-1821 |Transcendental | colspan="2" |Transcendental |- |1822-1842 |Gilded | colspan="2" |Gilded |- |1843-1859 |Progressive | colspan="2" |Progressive |- |1860-1867 | rowspan="3" |Missionary | colspan="2" rowspan="3" |Missionary |- |1868-1877 |明治一桁世代 (Meiji Hitoketa Sedai) |- |1878-1882 | rowspan="3" |明治二桁世代 (Meiji Futaketa Sedai) |- |1883-1900 |Lost | colspan="2" |Lost |- |1901-1911 | rowspan="2" |G.I. | colspan="2" rowspan="2" |G.I |- |1912-1924 | rowspan="2" |大正世代 (Taisho Sedai) |- |1925 | rowspan="6" |Builder | rowspan="5" |Silent | rowspan="2" |G.I ! rowspan="2" |Silent |- |1926-1927 | rowspan="2" |昭和一桁世代 (Showa Hitoketa Sedai) |- |1928-1934 | rowspan="4" |Silent | colspan="2" rowspan="3" |Silent |- |1935 |昭和一桁世代 (Showa Hitoketa Sedai) / 焼け跡世代 (Yakeato Sedai) |- |1936-1942 | rowspan="3" |焼け跡世代 (Yakeato Sedai) |- |1943-1945 | rowspan="4" |Baby Boom !Silent |Baby Boom |- |1946 | rowspan="4" |Baby Boom | rowspan="4" |Baby Boom | colspan="2" rowspan="3" |Baby Boom |- |1947-1950 |団塊世代 (Dankai "Boomer" Sedai) |- |1951-1960 | rowspan="2" |しらけ世代 (Shirake Sedai) |- |1961-1963 | rowspan="4" |Generation X !Baby Boom |Generation X |- |1964-1970 | rowspan="2" |Generation X | rowspan="3" |Generation X | colspan="2" rowspan="2" |Generation X |バブル世代 (Baburu "Bubble" Sedai) |- |1971-1979 | rowspan="2" |氷河期世代** (Hyougakki "Ice Age" Sedai) |- |1980-1981 | rowspan="3" |Generation Y !Generation X |Millennial |- |1982-1986 | rowspan="3" |Millennial | rowspan="4" |Millennial | colspan="2" rowspan="2" |Millennial |プレッシャー世代 (Puresshaa "Pressure" Sedai) |- |1987-1994 | rowspan="2" |ゆとり世代 (Yutori Sedai) |- |1995-1996 | rowspan="3" |Generation Z !Millennial |Generation Z |- |1997-2005 | rowspan="3" |Generation Z |Millennial !Generation Z |脱ゆとり・さとり世代 (Datsu Yutori / Satori Sedai) |- |2006-2009 | rowspan="4" |Homeland !Generation Z | | |- |2010-2011 | rowspan="2" |Generation Alpha |Generation Z !Generation Alpha | |- |2012-2024 | rowspan="3" |TBD | colspan="2" |Generation Alpha | |- |2025-2029 | rowspan="2" |Generation Beta | colspan="2" rowspan="2" |TBD | |- |2030-2039 |TBD | |} {| class="wikitable" !Generation !Strictest !Probable !Broadest |- |Arthurian |1433-1460 (28) |1433-1460 (28) |1433-1460 (28) |- |Humanist |1461-1482 (22) |1461-1482 (22) |1461-1482 (22) |- |Reformation |1483-1511 (29) |1483-1511 (29) |1483-1511 (29) |- |Reprisal |1512-1540 (29) |1512-1540 (29) |1512-1540 (29) |- |Elizabethan |1541-1565 (25) |1541-1565 (25) |1541-1565 (25) |- |Parliamentary |1566-1587 (22) |1566-1587 (22) |1566-1587 (22) |- |Puritan |1588-1617 (30) |1588-1617 (30) |1588-1617 (30) |- |Cavalier |1618-1647 (30) |1618-1647 (30) |1618-1647 (30) |- |Glorious |1648-1673 (26) |1648-1673 (26) |1648-1673 (26) |- |Enlightenment |1674-1700 (27) |1674-1700 (27) |1674-1700 (27) |- |Awakening |1701-1723 (23) |1701-1723 (23) |1701-1723 (23) |- |Liberty |1724-1741 (18) |1724-1741 (18) |1724-1741 (18) |- |Republican |1742-1766 (25) |1742-1766 (25) |1742-1766 (25) |- |Compromise |1767-1791 (25) |1767-1791 (25) |1767-1791 (25) |- |Transcendental |1792-1821 (30) |1792-1821 (30) |1792-1821 (30) |- |Gilded |1822-1842 (21) |1822-1842 (21) |1822-1842 (21) |- |Progressive |1843-1859 (17) |1843-1859 (17) |1843-1859 (17) |- |Missionary |1860-1882 (23) |1860-1882 (23) |1860-1882 (23) |- |Lost |1883-1900 (18) |1883-1900 (18) |1883-1900 (18) |- |G.I. |1901-1924 (24) |1901-1924 (24) |1901-1927 (27) |- |Silent |1925-1945 (21) |1925-1945 (21) |1928-1945 (18) |- |Baby Boom |1946-1963 (18) |1946-1963 (18) |1946-1963 (18) |- |Generation X |1964-1979 (16) |1964-1979 (16) |1964-1981 (18) |- |Millennial |1980-1994 (15) |1980-1995 (16) |1982-1996 (15) |- |Generation Z |1995-2009 (15) |1996-2011 (16) |1997-2011 (15) |- |Generation Alpha |2010-2024 (15) |2012-2024 (13) |2012-2024 (13) |- |Generation Beta |2025-2039 (15) |2025-? |2025-? |} == Notes == [a] Mccrindle's 'Builders' align best with other models' Silent Generation [b] Mccrindle's 'Generation Y' aligns best with other models' Millennial Generation [c] Whilst not explicit, any direct successor of Generation Z proposed by Pew Research Center must align best with other Mccrindle's 'Gen Alpha' and S&H's 'Homeland Generation' [d] S&H's 'Homeland Generation' aligns best with Mccrindle's Generation Alpha == References == https://mccrindle.com.au/article/topic/demographics/the-generations-defined/ https://www.pewresearch.org/short-reads/2019/01/17/where-millennials-end-and-generation-z-begins/ https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory https://www.genwhy.com/_files/ugd/5ccc25_695074dbc5724d76ae103635b5d9ab5d.pdf ayccvg9a7y7byyeo9gyhrxer47w55pp 2 317752 2835667 2694775 2026-10-03T10:24:52Z B.NIROSHAN 2980192 /* Information system */ 2835667 wikitext text/x-wiki == Computer system == consists of # hardware # software # data # user == Information system == 2odqcp89z5hjdqhdfl1fpt5dfz8jt9m 0 326060 2835502 2835491 2026-10-02T12:39:17Z Jeanne Noiraud 1366702 /* Conceptual modelling */ 2835502 wikitext text/x-wiki == Acknowledgements == === Contributors === {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling, article writing |- |Amélie E. Pereira |Laboratoire DICEN IDF |https://orcid.org/0009-0005-5928-5586 |Meta-data enrichement, article writing |- |Finn Nielsen |Technical University of Denmark |https://orcid.org/0000-0001-6128-3356 |Data visualisation |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice.<ref>{{Cite Q|Q134545572}}.</ref><ref>{{Cite Q|Q129947262}}.</ref> The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors.<ref name="Geels2020">{{Cite Q|Q121307039}}.</ref><ref name="GeelsAyoub2023">{{Cite Q|Q134545555}}.</ref><ref name="Sovacoolal2025">{{Cite Q|Q134545611}}.</ref> Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences.<ref>{{Cite Q|Q134545576}}.</ref><ref>{{Cite Q|Q46429136}}.</ref> Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination. === The problem of academic information overload === Global scientific output is quickly growing<ref>{{Cite Q|Q100712015}}.</ref><ref name="Hansonal2024">{{Cite Q|Q131700697}}.</ref><ref>{{Cite Q|Q141613068}}</ref>, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure<ref name="Hansonal2024" />. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge.<ref>{{Cite Q|Q134545560}}.</ref> “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily. === The problem of knowledge synthesis === The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable.<ref>{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}</ref><ref>{{Cite Q|Q55121944}}.</ref> and for shifting from static text format publications to dynamic knowledge mapping<ref name="Krlev2019">{{Cite Q|Q141608208}}</ref> This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : <ref>{{Cite Q|Q40040379}}.</ref><ref>{{Cite Q|Q134545630}}.</ref><ref name="Spadaroal2022">{{Cite Q|Q141606483}}.</ref>). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia. === Visualize the statements supported by a specific source === * https://synia.toolforge.org/#work/Q114306483 === The problem of scientific results dissemination === There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful.<ref>{{Cite Q|Q134545557}}.</ref> Wikipedia is a key knowledge dissemination platform widely used by students.<ref>{{Cite Q|Q118144142}}.</ref> and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature.<ref>{{Cite Q|Q42013239}}</ref> and that some scholars cite directly Wikipedia.<ref>{{Cite Q|Q66709114}}.</ref> However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives.<ref>{{Cite Q|Q130493308}}.</ref><ref>{{Cite Q|Q107303605}}.</ref>, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow<ref name="Wikimania2011survey">{{Cite Q|Q141607620}}.</ref> In addition, expert participation is not immune to the gender gap<ref name="Wikimania2011survey"/>. Because of gender segregation in disciplines<ref>{{Cite Q|Q28265069}}.</ref>, this may be detrimental to the content coverage on “female” topics<ref>{{Cite Q|Q27797938}}.</ref>, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]]. == Toward a living review on just sustainability transition == === Just sustainability transition === Just sustainability transition transition is "a fair and equitable process of moving towards a post-carbon society"<ref name="McCauleyHeffron2018">{{Cite Q|Q129947262}}.</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name="McCauleyHeffron2018"/>. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name="McCauleyHeffron2018"/>. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name="Xinxinal2021">{{Cite Q|Q137209041}}</ref><ref name="Heffron2021">{{Cite Q|Q134545572}}.</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name="McCauleyHeffron2018"/><ref name="Xinxinal2021"/><ref name="Heffron2021"/><ref name="Jenkinsal2016">{{Cite Q|Q137210566}}.</ref>. Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis<ref name="Geels2020" /><ref name="GeelsAyoub2023" /><ref name="Sovacoolal2025" /> and fragmented results coming from various disciplines<ref name="Droubial2022">{{Cite Q|Q137901182}}.</ref>. Then, making literature reviews "living" would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories. === Living reviews === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Aklal2017">{{Cite Q|Q50084143}}</ref><ref name="Simmondsal2021">{{Cite Q|Q141608286}}.</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1" /><ref name="Aklal2017"/><ref name="Simmondsal2021"/>. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Aklal2017"/><ref name="Simmondsal2021"/><ref name="Krlev2019"/>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name="Simmondsal2021"/>, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities<ref name="Krleval2025">{{Cite Q|Q141608155}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref name="Krleval2025"/>, as IA are useful to extract, filter and classify datas<ref>{{Cite Q|Q141608482}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but not yet integrated into tested and validated methodologies<ref name="Metzendorfal2025">{{Cite Q|Q134545593}}</ref>. Human validation stays notably necessary<ref>{{Cite Q|Q123252496}}.</ref><ref name="Metzendorfal2025"/>. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process. === Wikimedia projects === Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research<ref>{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&oldid=1360514388|journal=Wikipedia|language=en}}</ref>, we are working on two sister projects : Wikidata and Wikiversity. As its name suggests (wiki and data), [[wikipedia:Wikidata|Wikidata]] is, like Wikipedia, a collaborative project based on wiki editing technology; however, unlike the online encyclopaedia, it contains data rather than articles. Whilst there is a separate Wikipedia for each language version, there is only one Wikidata, containing items organised using unique identifiers. What matters when organising information in Wikidata is not the words used to define each concept, but the concept itself. Wikidata is a [[Databases|database]], more precisely a knowledge base, "collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation".<ref>{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}</ref><ref>{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&oldid=1360462340|journal=Wikipedia|language=en}}</ref> "A [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.<ref name="Hoganal2022">{{Cite Q|Q141609055}}.</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite Q|Q130027230}}.</ref> Such graphs have a strong potential to conduct knowledge synthesis.<ref name="Krlev2019"/><ref name="Fotopouloual2022">{{Cite Q|Q117837999}}</ref><ref name="Spadaroal2022"/> They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge.<ref name="Spadaroal2022"/> Wikidata is a unique project, designed from the outset for multilingual use.<ref name="VrandečićKrötzsch2014">{{Cite Q|Q18507561}}.</ref> Structuring the data in the form of RDF triples enables a multilingual understanding of the information, provided that their metadata (the ‘label’ and ‘description’ fields) are also translated.<ref name="VrandečićKrötzsch2014"/><ref>{{Cite Q|Q27044266}}</ref> Each Wikidata element in fact comprises a box linking a label (the common term used to refer to the element in each language) to a descrition. Integration into WD is based on adherence to the five principles of linked open data (available on the web, under a free licence, in RDF format, with a unique, identifiable URL for each item, and linked to other data sources), under a CC0 licence, which places the data in the public domain. The use of a unique identifier is a cornerstone of linked open data.<ref>{{Cite Q|Q115725978}}</ref> Indeed, in order to link two databases or knowledge bases together, it is necessary to use an identifier that is both unique to a record or a concept and common to both databases, so as to enable data sharing. This identifier is generally derived from an authority source. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]]).<ref>{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&oldid=1358552930|journal=Wikipedia|language=en}}</ref><ref>{{Cite Q|Q100300896}}.</ref> Like all Wikimedia projects, Wikiversity is editable by everyone, have a discussion tab and an history log tab. === Use of Wikidata and Wikiversity for research === A review specifically focusing on the uses of Wikidata in digital humanities projects was carried out by Fudie Zhao in 2022.<ref name="Zhao2022">{{Cite Q|Q116369826}}.</ref> It concluded that WD is used as a content provider, a platform and a technology stack. More specifically, these uses include the implementation of data annotation and enrichment, metadata curation, knowledge modelling and named entity recognition.<ref name="Zhao2022"/> Applications in the life sciences were analysed by Waagmeester et al. in 2019, who highlighted its alignment with the FAIR principles as one of its key strengths for this purpose<ref name="Waagmeesteral2020">{{Cite Q|Q87830400}}</ref>. In this field, WD has been enriched with content from authoritative public sources on genes, proteins and chemical compounds. WD is also used in the social sciences and in artificial intelligence research.<ref>{{Cite Q|Q118130164}}</ref> Wikidata is used as a crowdsourced curation of biomedical ontologies. These ontologies are essential for structuring biomedical knowledge, but their maintenance requires substantial ongoing effort, including updating content and mapping terms to other ontologies and vocabularies. Community curation can distribute this work among many contributors, although it may also introduce errors and inconsistencies. The study investigated Wikidata as a crowdsourced curation platform for the Human Disease Ontology (DO), combining community contributions with expert review. Disease data from DO were initially added to Wikidata in 2015, and from 2018 onward, Wikidata was compared monthly with the latest DO release. Beyond identifier mappings, Wikidata users proposed structural changes to the ontology, such as new disease terms and modifications to subclass relationships. These suggestions were less often incorporated directly, but frequently prompted expert review and refinement. The authors conclude that this hybrid crowdsourcing model can be applied to other resources synchronized with Wikidata, allowing distributed contributions while retaining expert control over the final curation. Wikidata also increases the visibility and accessibility of biomedical ontologies by connecting them with a broader community of users, tools, and services.<ref name="Waagmeesteral2020"/> Wikidata was relevant because it provides a collaborative platform for aggregating and integrating biomedical knowledge from a large community of contributors. The Human Disease Ontology (DO), meanwhile, provides a structured representation of disease concepts and their relationships. Combining the two made it possible to distribute part of the curation work among Wikidata users while retaining expert control over the quality of the ontology. The researchers used the Disease Ontology as a full ontology, rather than simply as a vocabulary list. The ontology contains disease concepts as well as relationships between them, such as subclass relationships, and mappings to external vocabularies and resources. Wikidata was used as a crowdsourcing and knowledge-integration platform. Contributors could add identifier mappings to resources such as GARD, MeSH, ORDO, and OMIM, as well as propose changes to the ontology structure and new disease terms. These contributions were then reviewed by expert curators before being incorporated into the official Disease Ontology. The researchers seeded Wikidata with disease items from the Disease Ontology and then developed a system to monitor, filter, and prioritize changes made by Wikidata contributors. From 2018 onward, they compared Wikidata with the latest official Disease Ontology release on a monthly basis. The community contributed new mappings and structural suggestions, while expert DO curators manually reviewed and validated these contributions. Most identifier mappings were accepted after review, whereas more complex structural changes were generally examined and refined by experts before being incorporated into the ontology. Thus, the ontology was developed through a hybrid process combining distributed community curation with expert validation.<ref name="Waagmeesteral2020"/> / Short version / Waagmeester et al. (2020)<ref name="Waagmeesteral2020"/> evaluated Wikidata as a crowdsourced curation platform for the Human Disease Ontology (DO) using a hybrid model that combines community contributions with expert review.<ref name="Waagmeesteral2020"/> Wikidata was seeded with DO disease items starting in late 2015, and from 2018 onward the disease data were compared monthly with the most recent DO release. In the first comparison, Wikidata users had added 2,030 cross-references to GARD and MeSH, mainly through a small number of contributors using an identifier-mapping interface. After manual review by DO curators, 2,007 (98.9%) were judged correct and included in the next DO release. Subsequent monthly reports, which also included ORDO and OMIM mappings, showed approval rates above 90%. Contributors also proposed structural changes, such as new disease terms and revised subclass relationships. These were rarely adopted without modification, but they often prompted further review and refinement by DO curators. The authors argue that this approach can be generalized to any resource synchronized with Wikidata, allowing distributed contributions while preserving expert control over the final curation. They also note that integration into Wikidata increases the resource's visibility by engaging a broader community of users, curators, tools, and services. === Knowledge graphs for research === The theory database of behaviour change (https://theory-database.hbcptools.org/) references more than 70 theories of behaviour change and uses an ontology-based modelling system (OBMS) to represent constructs and their relationships in a formalized way.<ref name="Westal2019">{{Cite Q|Q92955740}}.</ref><ref name="Haleal2020">{{Cite Q|Q102211870}}.</ref> Behaviour-change theories are written in natural language, sometimes with ad hoc diagrams. This makes them ambiguous: the same construct gets different labels, and the same label is used for different constructs, even within one theory. The authors argue this hinders comparing, integrating and applying theories, and may explain why interventions are often disconnected from explicit theory. A review of 83 theories showed these problems. Existing resources did not solve this. The Grid-Enabled Measures database, the Science of Behaviour Change repository and the Theoretical Domains Framework organise or synthesise constructs, but none standardises how theories are represented. A formal system of defined constructs and typed relationships, of the kind used to build ontologies, was chosen to make theories precise and comparable. It would also allow computational searching, clustering and reasoning across theories. It is closer to a full ontology than to a vocabulary list, though the authors do not claim a finished ontology. The system has three parts: * Defined constructs. Each has a label and a definition (85 constructs across the five theories). * Typed binary relationships. Each theory is written as "subject–relationship–object" propositions. The relationships fall into three types: causal (e.g. "is influenced (+) by", "may be influenced by", "correlates with"), semantic ("type of", "value of", "has attribute") and structural ("part of", "has start", "has end"). * Diagrams. Propositions are also shown graphically, and the diagrams can be exported as subject–relationship–object triples. The authors say such representations "can be used to create formal 'ontologies'". They note, though, that the system is not yet formally linked to OWL, which would allow greater expressivity and automated inference. They plan to do this in the next phase. They selected five theories from the 83 in Davis et al.: the Health Belief Model, the Information-Motivation-Behavioural Skills model, Social Cognitive Theory, the Theory of Planned Behaviour and the Transtheoretical Model. These were chosen as the most frequently used in behaviour-change interventions. Construct labels were taken from the original published descriptions. Definitions were drafted from the theory text and reviewed by two additional authors. Where the theory text gave no definition, they consulted the APA dictionary, then other dictionaries and sources. Failing that, they wrote definitions from the theory text and dictionary definitions of the label's words. The relationship library was built incrementally, adding types only as needed to capture the theories. Two researchers independently coded each theory into triples, then reached an agreed version through team discussion. The construct definitions and theory representations were sent to the theory authors or, where authors couldn't be reached, to experts. The experts were found by searching Web of Science and Scopus for the most frequent citers of each theory. Proposed changes were included, or the authors explained why not. Validation was qualitative. There was no quantitative measure such as inter-rater agreement. The authors report that the theory authors "appeared to be satisfied" that the system captured their theories, but they gave no formal test. The reviews also exposed issues. The Theory of Planned Behaviour author did not think the "behavioural belief" construct was needed, but the authors kept it because the logic requires it. The Transtheoretical Model author noted that the representation omits transitions back to earlier stages, which the system could in principle handle. Limits : Only five theories were covered. No formal link to OWL yet. The representations capture only the main parts of each theory, and the diagrams become hard to read as propositions are added. . Why was an ontology-based approach relevant? Natural language is a poor medium for theories. Behaviour-change theories are expressed in natural language, which the authors say is ambiguous and limits comparison, integration and testing. Overlapping scope, vague definitions and inconsistent construct labels make it hard to choose or combine theories. Theories need to be computable. To search, compare, test and integrate theories efficiently, they must be specified in a way that is "clear, consistent and computable". Ontology-style construct–relationship–construct triples make theories computer-readable and searchable, and give a basis for logical inference. No existing ontology covered this domain. A scoping review found 15 ontologies related to human behaviour change, none of which represented its breadth and detail. This motivated the project's own Behaviour Change Intervention Ontology (BCIO), which the authors plan to link the theories to. 2. Vocabulary list or full ontology? Neither exactly. The authors state that the OBMS "is not an ontology", because ontologies typically represent knowledge about the world rather than theories about it. It does use the same modelling approach that ontologies use. What it is. It is a formal system of labelled constructs plus typed relationships, written as triples such as "intention–influences–behaviour". Diagrams are drawn in LucidChart and exported as CSV, and Python scripts extract the triples into a searchable database (Flask web interface, Whoosh search). Relationship types. 14 were used across the 76 theories. The most common were "influences" (995 triples) and "part of" (578), followed by "type of", multiplicative and positive influences, and others. More than a vocabulary list. It goes beyond a flat list of terms because it encodes relationships between constructs, including causal, semantic (type of), structural (part of) and process ("transitions to") types. Moderators are handled by "reification", treating a relationship as a construct. Not yet fully ontological. Constructs are identified by their labels only. There are no definitions or formal semantics yet, and this is the main gap. The authors plan to annotate constructs with entities from the BCIO, which would add synonyms, definitions and formal logic (e.g. OWL) to check consistency. The only formal ontology named in the paper is the BCIO, and it is a planned future link rather than something used here. 3. How was it developed and validated? Development Selection. 83 theories were screened from Davis et al. (2015). 71 new ones were represented, and combined with the five from West et al. (2019) they give 76. Seven were excluded: two did not link any construct to behaviour, one lacked a behaviour construct, and four had been integrated into the Theory of Triadic Influence. Constructs. At least two researchers independently extracted constructs from the original sources. Disagreements were settled by discussion, or by the study leads (SM and RW) if needed. The extraction was then cross-checked against the theory summaries in Michie et al. (2014). Diagrams. One researcher drafted each diagram, and a second or third refined it. There were 19 conceivable relationship types at the start, revised for parsimony. Only one type ("transitions to") was added to those in West et al. (2019). Validation Author and expert review. Each diagram went to the theory author or an expert. If none of the 15 authors could be reached (14 deceased, 1 retired), an expert was found through Web of Science and Scopus. Of the 69 contacted, 50 (72.5%) responded and 38 gave feedback. Iterative revision. A researcher identified changes and a second reviewed them, with revisions repeated until agreement. A non-response after two reminders counted as approval. Outcome. 35 theories (49%) were finalised with no or very minor changes. The other 36 (51%) changed constructs (7), relationships (15) or both (14). Openness. Data, diagrams and code are on OSF, and the database is online. Limitations and reviewer concerns Non-response counted as approval, and validation was qualitative with no agreement statistics. Only about 20% of construct names appeared in more than one theory. Both reviewers suspect this understates conceptual overlap, because constructs were matched by label rather than definition. Reification makes triples nested and complex, and the reviewers question how well moderators are represented. The reviewers also asked how the system handles non-deliberative, environmental (behaviourist) models and socio-cultural influences. Persons in Context (PiCo) is a knowledge model allowing to link the reconstruction of historical persons to observations and sources<ref>{{Cite Q|Q141609117}}</ref>. == Research question and hypothesis == Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?''' In this project, we aim to test 4 hypothesis : ● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations. ● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…). ● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs. ● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data). We also have 2 assumptions : ● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool. ● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers. == Methodology == Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy. == 1. Building an academic corpus and enriching bibliographic metadata == The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus. === Database search === Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions<ref name="Jenkinsal2016"/> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite Q|Q137210229}}</ref> (e.g. the participation of affected communities in decisions related to the construction of new infrastructures<ref name="McCauleyHeffron2018"/>). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)<ref>{{Cite Q|Q137210566}}</ref>. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions). We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article selection === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors * Including justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974 === Importing selected articles into Wikidata === Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through metadata enrichement === Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. <ref name=":5" />). By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}. ==== Adding {{Wikidata entity link|P921}} ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Adding {{Wikidata entity link|P8363}} ==== The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}.<ref>{{Cite Q|Q115443776}}</ref> After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |theory-driven evaluation used in evaluating social programmes |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |study of the statistical properties of combinations of studies from a meta-analytic dataset |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Adding {{Wikidata entity link|P6153}} ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Adding {{Wikidata entity link|P50}} ==== When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages. ==== Adding {{Wikidata entity link|P6977}} ==== We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews. === Advantages and limitations of Wikidata to build a rich living academic corpus === To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata. The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis. ==== Advantages of Wikidata ==== Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, "energy transition" was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). ==== Limitations of Wikidata ==== The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases). Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to "filter" the part of the knowledge graph we were working on. Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data. ==== Future possbilities ==== A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :<ref>{{Cite Q|Q118139086}}</ref>) Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms. == 2.Modelling the content of litterature reviews == The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations. === Concept mapping, thematic networks and causal networks === This section presents social science methodology that presents similarities with knowledge modelling. ==== Concept maps ==== [[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)<ref name="Cañasal2004">{{Cite Q|Q141613640}}</ref>. Concept maps can be a powerful literature review tool<ref>{{Cite Q|Q134545591}}</ref> allowing to synthetize theoretical statements about relationship between concepts.<ref>{{Cite Q|Q30884964}}</ref> They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]<ref name="Cañasal2004"/>, Altas.Ti "network" feature...). The "box and arrow" logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name="Attride-Stirling2001">{{Cite Q|Q134545550}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite Q|Q140050249}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name="Attride-Stirling2001"/>. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. [[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]] ==== Causal diagrams ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite Q|Q141622061}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref name="Gioiaal2013">{{Cite Q|Q141609039}}</ref> After identifying themes, qualitative researchers are expected to theorize the "arrows" between themes<ref name="Gioiaal2013"/> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other).<ref>{{Cite Q|Q141609011}}</ref> === Knowledge modelling in Wikidata : first round of analysis === This section presents our knowledge modelling experimentation in Wikidata. ==== Conceptual modelling ==== Our first step was to reflect on what is a "concept" and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts<ref name="Podsakoffal2016">{{Cite Q|Q141609106}}</ref> and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...<ref name="Podsakoffal2016"/>, and scholars have advised to take insight from philosophy to work on concepts<ref>{{Cite Q|Q141609085}}</ref>. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts.{{Cite Q|Q141622252}} We summarize these approaches below and examine which wikidata properties exist to represent them. *Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... *Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties). *Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en). *Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be. *Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==== To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : *{{Wikidata entity link|Q137901202}} *{{Wikidata entity link|Q137901196}} *{{Wikidata entity link|Q137901182}} *{{Wikidata entity link|Q136447761}} *{{Wikidata entity link|Q129652515}} *{{Wikidata entity link|Q114306483}} We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated "Energy democracy is both an ideal and a process"<ref>{{Cite Q|Q137901182}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. [[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]] We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process. ===== Contradictions ===== Wikidata contributor's feedback highlighted some apparent contradictions (The values in "does not have effect" seems contrary to what is listed in "has goal".) We would however argue this is not a problem because "statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata<ref name="DoğanPatel-Schneider2025">{{Cite Q|Q141613019}}</ref>, as long as they are supported by references.<ref>{{Cite Q|Q112038343}}</ref> Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, some authors specifically critique the discrepancy between the movement's stated goals and what it actually achieves.<ref name="Droubial2022"/> Recommandations : Contradictions are allowed in Wikidata. ===== Precision ===== We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite Q|Q129652515}}.</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata. [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between "concept" and "political concept".]] ===== Concision ===== Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were "in the tree of another value". The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification : *We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity<nowiki/>) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using "instance of" as transversal property and checking "instance of or subclass of"). *We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item. Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. ===== Quantification ===== Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources".<ref>{{Cite Q|Q137901196}}.</ref> Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for "final energy consumption". There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible, nor desirable. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). [[File:Wikidata screenshot of percentage modelling.png|border|783x783px]] Since Wikidata is optimized for statements with a few discrete values, it is also more convenient to store quantitative (tabular) data elsewhere<ref>{{Cite journal|last=Larrañaga|first=Galder Gonzalez|last2=Pintscher|first2=Lydia|last3=Ainali|first3=Jan|last4=Fauconnier|first4=Sandra|date=July 2026|title=Tabular data vs structured data.pptx|url=https://docs.google.com/presentation/d/1k37UX-VPwMgJbXoB7ZxsztE5LdKR_isM|journal=Wikimania 2026}}.</ref>. It is also possible to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. Since our aim is not to display or analyse quantitative data, we opted for this last solution. Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured and link it toward an external datasets containing the quantitative data. ===== Ontology ===== Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion "{{Wikidata entity link|Q127270577}}"). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea. Our first attempt to multiple "natures" of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science. For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)<ref>{{Cite Q|Q141612935}}.</ref>. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. In theory, such ontology can be implemented in Wikidata as it "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name="DoğanPatel-Schneider2025"/> Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). {| class="wikitable" |+ !Item !Existing root class entity |- |{{Wikidata entity link|Q190539}} |{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q67518978}} |- |{{Wikidata entity link|Q34770}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q9081}} |{{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q43229}} |{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q103940464}} |- |{{Wikidata entity link|Q49773}} |{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}} {{Wikidata entity link|Q99527517}} {{Wikidata entity link|Q30241068}} {{Wikidata entity link|Q7048977}} |- |{{Wikidata entity link|Q12705}} |{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}} {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}} {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}} |} Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. We discussed with the Wikidata community<ref>{{Cite web|url=https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology|title=Wikidata talk:WikiProject Ontology - Wikidata|website=www.wikidata.org|language=en|access-date=2026-07-28}}</ref> and explored existing ontologies that could be relevant to our project. Some colleagues social science and humanities recommended CIDOC-CRM (https://cidoc-crm.org/crmsoc/fm_releases) because of the active community maintaining it. The conclusion of this exploration is that Wikidata, because of its collaborative nature, may not be the best plateform to implement a rigorous, interoperable, high-level ontology. We thus decided to focus on a lower ontological level. === Knowledge modelling in Wikidata : second round of analysis === Our first round of analysis resulted in : * Add links to exported files + insert graphs screenshots After our first round of analysis, we decided to focus on the strenghs of wikidata, which seem to be its ability to be a hub for concepts linked to external identifiers. We reframed our modelling goals to focus on identifying the main concepts in our corpus, their definition, their relationships, and their potential existing identifiers. Our next steps are * Read articles of the corpus again and model more precise statements * Include pages in references * Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies. We started by relying on {{Wikidata entity link|Q136447761}}, which states more clearly the different facets of {{Wikidata entity link|Q14944319}}. We kept the first item {{Wikidata entity link|Q14944319}} as the main conceptual entity, representing the ideal of energy democracy as a goal. We created other items such as {{Wikidata entity link|Q141224129}} to represent the social movement advocating for energy democracy, {{Wikidata entity link|Q141224695}} to represent policies havin energy democracy as goal, {{Wikidata entity link|Q141223998}} to represent the scholarly study of energy democracy (to separate ''phenomenon versus theory)'', {{Wikidata entity link|Q141224432}} to represent energy democracy as an outcome (to separate ''process versus outcome).'' Creating these different facets of {{Wikidata entity link|Q14944319}} allowed more precise modelling. Then existing statements were sorted to match the different types of entity. For example, {{Wikidata entity link|Q3109572}} was linked to {{Wikidata entity link|Q141224129}} and {{Wikidata entity link|Q1951366}} to {{Wikidata entity link|Q141224695}}. To reflect the dimensions of the concept of {{Wikidata entity link|Q14944319}}, we used {{Wikidata entity link|P527}}. For example, the sentence "Becker and Naumann list decentralized energy generation, public and cooperative ownership and energy sovereignty as the dimensions of energy democracy<ref>{{Cite Q|Q139492906}}</ref><ref>{{Cite Q|Q136447761}}</ref>, was modeled as {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P527}} : {{Wikidata entity link|Q861135}}, {{Wikidata entity link|Q119080352}}, {{Wikidata entity link|Q4227971}} and {{Wikidata entity link|Q130487674}}. We removed some statements, following logical simplifications. For example, there is no need to mention the broad concept of {{Wikidata entity link|Q6498684}}, while there are mentions of more precise concepts such as {{Wikidata entity link|Q119080352}} and {{Wikidata entity link|Q4227971}}. == 3. Data visualisation == The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below. === Filter statements === *Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Visualize the statements supported by a specific source === * https://synia.toolforge.org/#work/Q114306483 === Properties visualisation === *Visualise the "tree" of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io/wikidata-graph-builder/) [[File:Visualisation of the "instance of" values for energy democracy in Wikidata.png|center|thumb|553x553px|Visualisation of the "{{Wikidata entity link|P31}}" values for {{Wikidata entity link|Q14944319}} in Wikidata. Made with the [https://angryloki.github.io/wikidata-graph-builder/ Wikidata Graph Builder].]] === Concepts visualisation === *Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request "topic in context")[[File:Screenshot of Energy democracy wikidata graph.png|center|thumb|450x450px|Graph of the {{Wikidata entity link|Q14944319}} concept generated with [https://scholia.toolforge.org Scholia]'s "The topic in context" query.]]Map the statements that two items have in common to make comparisons. [[File:Community energy and energy democracy.png|center|thumb|440x440px|Data visualisation of the concepts of {{Wikidata entity link|Q14944319}} and {{Wikidata entity link|Q113514984}}. Made with [https://think.resoneo.com/wikidata-search/ Wikidata Knowledge Explorer] .]]Find the relationships between two concepts. === Mapping sources consensus === *Queries can help visualize which concept is central across sources : https://query.wikidata.org/#%23defaultView%3AGraph%0ASELECT%20%0A%20%20%3Fvalue%20%3FvalueLabel%20%3Fpublication%0A%0AWHERE%20%7B%0A%20%20wd%3AQ14944319%20%3Fp%20%3Fobject_statement%20.%0A%20%20%3Fobject_statement%20prov%3AwasDerivedFrom%20%3Freference%20%3B%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3Fps%20%20%3Fvalue%20.%0A%20%20%3Fvalue%20rdfs%3Alabel%20%3FvalueLabel%20.%0A%20%20FILTER%20%28lang%28%3FvalueLabel%29%20%3D%20%27en%27%29%0A%20%20%3Freference%20pr%3AP248%20%3Fpublication%20.%0A%7D[[File:Link between concepts and references in the Energy Democracy wikidata item.png|center|thumb|396x396px|Link between concepts and references in the Energy Democracy wikidata item.]] *Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.[[File:Number of references supporting statements about a chain of contributing factors of energy democracy.png|center|thumb|454x454px|Visualisation of the number of references supporting statements about the chain of contributing factors of energy democracy.]] == 4. Writing == The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). * To do idea : write a Wikiversity page for each central concept (e.g. energy democracy, energy community, energy citizenship...) === Advantages of Wikiversity === Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication).<ref>{{Cite Q|Q110684311}}</ref> The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way. Some researchers experimented relationships between the wikimedia community and researchers by making researchers review wikipedia articles.<ref>{{Cite Q|Q136659684}}.</ref> === The issue of text interoperability === A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format). There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish "remixed" texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community. == Discussion == Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion.<ref>{{Cite Q|Q141609095}}.</ref> Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to "navigate" it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to "navigate" scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes. A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph.<ref name="Hoganal2022"/> === Quality assessment === <ref>{{Cite Q|Q141622289}}</ref> Methods for assessing Wikidata's data quality are still under development: following the questions raised by Piscopo and Simperl about the accuracy and reliability of the data<ref>{{Cite Q|Q66776851}}.</ref>, Shenov et al. (2022) propose a framework combining several indicators to detect low-quality statements, revealing issues such as duplicate entities, missing triples, and unmet constraints.<ref>{{Cite Q|Q107425133}}</ref> Two studies address the quality of ''referencing'' in Wikidata — that is, the reliability of sources cited to support claims — a distinct but complementary aspect of data quality itself. Amaral et al. (2021)<ref>{{Cite Q|Q112038343}}.</ref> use a mixed-methods approach (crowdsourcing, statistics, machine learning) to assess the relevance, accessibility, and authority of Wikidata's references at scale and across languages. Beghaeiraveri et al. (2023) build on this by proposing the RQSS, a system that quantifies referencing quality through scores, allowing contributors and project leads to identify and fix gaps.<ref>{{Cite Q|Q130478432}}</ref> Together, these works mark a shift from exploratory assessment of reference quality toward standardized measurement tools — though this raises the question of whether a numerical score can adequately capture a notion as contextual as a source's "authority". The community develops tools for detecting vandalism (intentional additions of low-quality contributions), relying on machine learning approaches adapted to the specific features of RDF triples, up to the current deployment of the ORES service in Wikidata.<ref>{{Cite Q|Q27579144}}.</ref><ref>{{Cite Q|Q28942543}}</ref> === Future research === - Can we "reason" with Wikidata (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ? - Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html ==== Supporting inductive research ==== Inductive research with an empirical phenomenon and then use of abstract concepts to describe it, a process called “Framing”, theorizing or conceptualizing.<ref name="CasanaveLi2015">{{cite Q|Q141609034}}</ref> Conceptualization require to answer the question « what are our data examples of? ».<ref name="CasanaveLi2015"/> This step is particularly challenging in inductive research since the same empirical phenomenon can be described through different concepts that leads to different perspectives or lenses the phenomenon and will refer to different theoretical traditions or disciplines.<ref name="CasanaveLi2015"/> Building a collection of concepts and their relationship to specific concrete cases in Wikidata could support inductive process by giving researchers access to relevant conceptualizations made by others. == Funding == This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603. == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} g254jigvfxlwdh2n7pzzutu1x52vx78 0 327555 2835668 2835412 2026-10-03T10:39:47Z DavidMCEddy 218607 /* Key claims */ most important ... 2835668 wikitext text/x-wiki [[File:Pharoah - James VI and I - Trump.png|thumb|Religious and media leaders from the time of the Pharaohs convinced common folk to give increasing shares of what they produced to elites.]] :''This book uses dates in [[:w:ISO 8601|ISO 8601]], YYYY-MM-DD, when convenient.'' == Invitation to edit this book == You, dear reader, are invited to contribute questions, ideas and citations to support or refute claims made in this book possibly adding chapters. Wikiversity like other Wikimedia Foundation Projects invites humans to [[w:Wikipedia:Be bold|“be bold but not reckless,”]] while writing from a [[Wikiversity:Disclosures|neutral point of view]], [[Wikiversity:Cite sources|citing credible sources]]. Others are invited to change or revert what you wrote. What stays tends to be written from a neutral point of view citing credible sources. If someone reverts your edit or you have a question, take it to the ''[[Wikiversity:FAQ|''''“Discuss”'''' page]]'' associated with the specific Wikiversity page most related to your concerns. Those who teach media literacy are encouraged to invite their students to debate and revise the contents of this book. Doing so would build on a tradition of [[:w:Wikipedia:Student assignments|instructors requiring students to edit wikipedia article(s).]] Editing [[:w:Wikipedia|Wikipedia]] and other [[:w:Wikimedia Foundation|Wikimedia Foundation]] projects like this book is itself an exercise in media literacy: :''Central tenets of media literacy might include writing from a neutral point of view citing credible sources and engaging others, some of whom may disagree, in civil, supportive conversations about what can and cannot be said based on a reasonable evaluation of the available evidence. Wikimedia rules invite contributors to do just that, encouraging them to “be bold but not reckless,” contributing revisions written from a neutral point of view, citing credible sources -- and raising other questions and concerns on the ''''“Discuss”'''' page associated with the specific Wikiversity page most related to your concerns, as mentioned above.''<ref>For more on this, see Graves (2024).</ref> == Text and self-help book and point of discuss == This book is intended both as a text and self-help book and as a point of discussion considering four levels of media literacy: :1. '''Think before you share''': Follow the money. Every media organization sells changes in audience behaviors to those who control much of their money. Major media everywhere promote ''entrepreneurs of hate''<ref>Mokyr (2017, p. 60).</ref> to convince the public to give increasing shares of what they produce to elites to "protect them from designated evil others." To counter this, DiResta (2024, p. 335) recommends, "Think before you share." :2. '''Look for information to contradict preconceptions''' (Disconfirmation bias): [[w:Information is a public good: Designing experiments to improve government#Previous research|Virtually everyone]] (a) thinks they know more than they do ([[w:Overconfidence effect|overconfidence effect]]), and (b) prefers information and sources consistent with preconceptions ([[w:Confirmation bias|confirmation bias]]). The major media everywhere exploit this to please those who control most of the money for the media. Humans can counter this by searching for sources to help us understand our designated enemies. ''If we cannot explain circumstances under which we could see ourselves doing what we see our designated enemies doing, we have not looked hard enough.'' :3. '''Talk''': Push ourselves to have friendly supportive conversations with others with whom we may vehemently disagree, seeking common ground, agreeing to disagree agreeably where we differ, and building collaboration on areas of common concern.<ref>Graves and Bailey (2025).</ref> :4. '''Teach''': Humans who develop skills in the first three levels can leverage that knowledge in helping others acquire those skills. If each one teaches two<ref>"[[:w:Each one teach one|Each one teach one]]" is an African-American proverb from the time of legalized slavery. However, if each one teaches only one, the growth in literacy will only be linear. Having "each one teaching two", on average, unleashes the power of doubling and [[:w:exponential growth|exponential growth]], which has the potential of educating the entirety of humanity in a reasonable period of time -- namely after 33 doublings starting from one.</ref> in a certain period of time, that time period becomes a [[:w:Doubling time|doubling time]]. Ten doublings is a thousand -- actually 1,024 to be precise.<ref>2 time 2 = 4 times 2 = 8 times 2 = 16 times 2 = 32 times 2 = 64 times 2 = 128 times 2 = 256 times 2 = 512 times 2 = 1024: That's 10 doublings, as anyone with a modest understanding of modern digital [[:w:computer|computer]]s will tell you.</ref> Twenty doublings become a million. Thirty doublings become a billion. Three more doublings become 8 billion, the [[:w:World population|world population]] as of approximately 2022-11-15.<ref>This book uses dates in [[:w:ISO 8601|ISO 8601]], YYYY-MM-DD, when convenient.</ref> Many organizations, including several United Nations agencies and the [[Yonty Friesem on media education|Media Education Lab]], have been working on [[w:media literacy|media literacy]] for decades and have trained many.<ref>''[[Wikibooks:Antiracist Activism for Teachers and Students]]'' includes a chapter on [[Wikibooks:Antiracist Activism for Teachers and Students/Points to Consider for Teaching Anti-racism/Media Literacy In Schools|Media Literacy In Schools]].</ref> This book is being written hoping to increase the effectiveness and accelerate the rate of growth in media literacy and thereby accelerate progress against many of the most pressing issues facing humanity today.<ref>[[w:Nolan Higdon|Nolan Higdon]] and Spencer Graves discussed marketing media literacy in a "[[:Category:Media reform to improve democracy|Media & Democracy]]" episode on, "[[Media literacy and why you should care]]".</ref> Much of this book is a [[w:Monograph|research monograph]] summarizing research that seems to have been underreported by the major media to avoid offending people who control most of the money for the media. These research results seem to be central to major political divisions. Each chapter ends in exercises to help the reader practice media literacy skills and have fun doing it. Remember: :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.'' Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term "cockamamie" is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue. ''Never say, "You're wrong." Instead, ask, "May I offer a contrary perspective?" Or "May I share with you another view that I've heard?"'' Much of the information in this book seems to have been largely overlooked and perhaps suppressed, apparently because it would either offend people who control substantial portions of the money for the media our would increase the cost of producing news; see the brief discussion of conflicts of interest by the major media in the next "Key claims" section. ==Key claims== * ''Primary drivers of every major conflict include differences between the media or difference in interpretation that the different parties find credible''. :-- This works, because everything we think we know is coded in systems of connections between neurons in our brains. These systems are more unique than fingerprints and evolve over time. The words we use do not mean the same to two different humans nor even to the same human at different points in time. In many cases these differences are inconsequential. ''Sometimes they are fatal.''<ref>Graves and Bailey (2026).</ref> :-- ''[[w:Social constructionism|Show me someone who knows the truth]], and I will show you someone who is dangerous'' -- especially during war or any other situation where humans may be moved to violence encouraged by their belief system.<ref>[[w:Collateral damage|Collateral damage]] that "they" commit proves to "us" that "they" are subhuman or at best criminally misled and must be resisted by any means necessary. By contrast, collateral damage that "we" commit is unfortunate but necessary.</ref> * ''The major media everywhere have [[w:Conflict of interest|conflicts of interest ]] in honestly reporting on [[v:Information is a public good per communications prof Pickard|anything that might offend anyone who controls large portions of the money for the media]].''<ref>Pickard and Graves (2025), accessed 2026-02-08; Pickard (2020).</ref> :--The most important things you need to know to be a responsible citizen will rarely if ever appear in the major media. :--[[v:Media Reform Coalition challenges anti-democratic media bias in the UK|British journalist and media reform advocate Dan Hind]] said that the content produced by the [[w:BBC|BBC]] was frivolous, soap opera stuff, because leading media personalities know very little about issues of substance and believe "they might get in trouble if" they produced anything serious. Similar analyses seem to apply to the major media everywhere<ref>Hind and Graves (2025), accessed 2026-02-09.</ref> but may not apply to non-profit and local media, which seem more likely to produce [[w:Investigative journalism|investigative]] / [[v:Dean Starkman and the watchdog that didn't bark|accountability journalism]]:<ref>Usher and Kim-Leffingwell (2022); see also Starkman and Graves (2025), accessed 2026-02-09.</ref> [[w:Watchdog journalism|Watchdogs]] tend to protect the people who feed them. Argentine journalist [[w:Horacio Verbitsky|Horacio Verbitsky]] said, "Journalism is disseminating information that someone does not want known; the rest is [[w:propaganda|propaganda]]."<ref>p. 16 in Verbitsky (1997); English translation from [[Wikiquote:Horacio Verbitsky]], accessed 2026-02-09.</ref> * ''The major media everywhere create the stage upon which politicians read their lines.'' :-- Major media promote ''entrepreneurs of hate''<ref>Mokyr (2017, p. 60).</ref> to make it easier for elites to pick everyone else's pockets. This is particularly effective with for-profit social media, which make money herding their audience into echo chambers, where erroneous preconceptions are reinforced, increasing political polarization and violence. [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen said]], "The shortest path to a click is anger or hate." A leading expert on for-profit social media is Renée DiResta,<ref>DiResta (2024).</ref> whose research documents how lies spread on social media threaten public health and prepped a small army of humans throughout the 2020 US presidential election cycle to believe that there were major deficiencies in the integrity of US election procedures. When Trump lost, he was able to recruit [[w:January 6 United States Capitol attack|a crowd to attack the capitol on January 6, 2021]]. Social media influencers made so much money spreading lies that it seemed profitable for them to file frivolous lawsuits ([[w:Strategic lawsuit against public participation|SLAPP]]), hoping that DiResta's legal bills would convince her to stop trying to expose their lies. After Republicans regained control of the US House in 2022, they subpoenaed her, claiming she had taken orders from the "[[w:Deep state|deep state]]" to tell Facebook to censor conservatives. [[Managing the good and the evils of social media|When interviewed for ''Media & Democracy'', she reported legal bills of $7 million]]. :-- The social psychology behind this phenomenon exploits especially [[w:confirmation bias|confirmation bias]]. It has contributed to [[Media Literacy and You/Media consolidation, social media, and political polarization|the dramatic increase in political polarization and violence worldwide]], especially since the end of the [[w:Fairness doctrine|Fairness doctrine]] in 1987. :-- Democracy may be best protected by local news nonprofits, managed jointly by journalists and boards selected at random from registered voters, according to Pickard (2023), discussed in the section below on "[[Media Literacy and You#Government subsidies for news|Government subsidies for news]]". :-- Their selection of acceptable topics for news and entertainment create and maintain the "[[w:Overton window|Overton window]]", which is the range of acceptable political discourse. For example, in early 1964, US President [[w:Lyndon B. Johnson|Lyndon Johnson]] understood that he could lose the 1964 presidential election that year if he were seen to be soft on communism. His response was to clandestinely provoke an attack on US naval vessels in the Gulf of Tonkin, which he could then denounce as "unprovoked". During a dark and stormy night 1964-08-04 the [[w:USS Maddox (DD-731)|USS ''Maddox'']] and [[w:USS Turner Joy|''Turner Joy'']] spent a couple of hours "defending themselves" against radar snow, then [[w:Gulf of Tonkin incident|reported that they had sunk two attacking North Vietnamese torpedo boats]]; subsequent investigations found no evidence of the reported attacks. That incident was used to justify the [[w:Gulf of Tonkin Resolution|Gulf of Tonkin Resolution]], with only two dissenting votes in the US Congress: Those two dissenters were defeated in their next reelection campaigns, illustrating the point that ::''the major media create the environment in which many politicians cannot get elected without betraying the nation.'' :-- Nick Hart,<ref name=Hart><!--Nick Hart-->{{cite Q|Q135663983}}</ref> President and CEO of the Data Foundation,<ref><!--Data Foundation-->{{cite Q|Q134705118}}</ref> noted that President Trump in his first term signed the [[w:Foundations for Evidence-Based Policymaking Act|Foundations for Evidence-Based Policymaking Act]] of 2019, which was bipartisan legislation ostensibly mandating evidence-informed public policy.<ref>Hart mentioned the [[w:Foundations for Evidence-Based Policymaking Act|Foundations for Evidence-Based Policymaking Act]] when he was interviewed for [[Evidence-informed public policy|"Media & Democracy" 2025-07-31]].</ref> Tragically, the record since 2019 suggests that, ::''The US Congress is rarely allowed to consider solid research suppressed by the major media.'' * ''The development of technology is never neutral in its impact on inequality but is driven to benefit people with power.''<ref>Acemoglu and Johnson (2023)</ref>. :-- [[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|Improvements in agricultural technology from pre-history to the time of King James of the King James bible were managed to benefit elites.]] Then pamphlets and newspapers began to appear, the head of state stopped granting as many monopolies, and commoners began getting permission to become entrepreneurs. That transformed economic stagnation into growth in GDP per capita adjusted for inflation, initiating the [[w:Industrial Revolution|Industrial Revolution]], as documented in the chapter below on [[/The impact of the media on political economy since the time of the Pharaohs/]]. It also led to increasing inequality until organized labor got enough political power to demand and get a bigger share of the fruits of their labors. :--[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|In the US those benefits peaked during the Great Depression]], when the standard conservative mantra that blames the poor for their poverty did not sell newspapers. US President Franklin Roosevelt taxed the ultra-wealthy like they had never been taxed before or since, and dramatically reduced inequality, which continued to decline until increasing concentration of ownership of the media ushered in a new era of increasing inequality starting with the presidency of Ronald Reagan. :-- [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future#Role of the media|Acemoglu and Johnson insist that technology, including artificial intelligence, can be developed to benefit all. However, but it will not happen without action by the poor and middle class as follows]]: :# Alter the narrative, :# Build countervailing powers [like organized labor], and :# Develop technical, regulatory, and policy solutions to tackle specific aspects of technology’s social bias.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> :-- For more on this, see, e.g., the chapter on [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]]. === The value of noncommercial news outlets === Some of the problems with the media and their contributions to increasing political polarization and violence are documented in the research summary on "[[Information is a public good: Designing experiments to improve government]]" and in the podcast series available on Wikiversity under "[[:Category:Media reform to improve democracy]]" with leading experts discussing their recommendations. One of the most compelling of the references discussed in that podcast series is Usher and Kim-Leffingwell (2022), who tallied all the federal prosecutions for political corruption in each of the 94 [[w:United States federal judicial district|US federal court district]]s between 2003 and 2019. During that period, the number of journalists in the US fell by a factor of roughly 3 -- between 60 and 70 percent. They found no statistically significant impact on federal prosecutions for political corruption of that decline in the number of journalists. However, each member of the [[w:Institute for Nonprofit News|Institute for Nonprofit News]] (INN) in a federal court district in one year was associated with on average 1.4 additional prosecutions for political corruption the following year. Since federal prosecutions for political corruption averaged roughly 10 per federal court district per year during that period, each member of INN in a federal court district one year was associated with a 14 percent increase in federal prosecutions the following year. This suggests that the major media outlets that had so dramatically reduced their staffs had not substantively reduced the amount of investigative journalism they did. If we assume that the people prosecuted for political corruption also control substantive advertising budgets, then the major media outlets have conflicts of interest in honestly reporting on such. They may report on it if some other organization like a member of INN does the research and they are threatened with a loss of audience from not reporting on it. :'''''Major point''''': You and I benefit, the vast majority of humans on earth benefit, from news reports presumably published by members of INN that contributed to those on average 1.4 additional prosecutions for political corruption estimated by Usher and Kim-Leffingwell (2022). We benefit even if we never heard about the news reports that contributed to those prosecutions. We benefit even if we have never heard of the news outlets that presumably did the investigative journalism behind those additional prosecutions. Why? Because on average those news reports likely deterred other incidents of political corruption, which likely contributed to broadly shared economic growth and the development of new technology that ultimately benefit the vast majority of humanity. Other aspects of this are documented in the research on the impact of [[w:news desert|news desert]]s, which we summarize next. === Costs increase in news deserts=== There's a growing body of research describing what happens when local newspapers die. Perhaps most important, a 2018 research report by Gao et al. reported that the death of a local newspaper was followed by … increases in local tax revenue, averaging $85 per human per year.<ref name = Gao2018>Gao et al. (2018).</ref> That $85 was roughly 13 hundredths of a percent of the 2019 US GDP. That's mentioned in the 2025-07-17 interview with [[Democratic delusions: Fix the media to fix democracy|Natalie Fenton about her new book, ''Democratic Delusions, How the Media Hollows out democracy and What We Can Do About It'']]. One of the most spectacular example of the cost of a news desert is the [[w:City of Bell scandal|Scandal of Bell, California]]. Their local newspaper died around 1999. Roughly a decade later the city was nearly bankrupt in spite of having property tax rates among the highest in the nation. An investigation by the ''[[w:Los Angeles Times|Los Angeles Times]]'' documented that the city manager had a compensation package worth $1.5 million a year, well over double that of the President of the United States. Other senior city officials were similarly well-remunerated. Some of the city officials went to jail over that. Did the city manager decide after 1999, "Wow: The watchdog is dead. Let's have a party"? Malfeasance also increases in business as pollution and workplace accidents increase as does the cost of capital, because investors know their money is not as secure without a local newspaper. That leads to a reduction in investments in new products, services and processes -- slowing economic growth. See "[[Local newspapers limit malfeasance]]", esp. Kim et al. (2021). And executive compensation in increases in nonprofits, so less of what people donate goes to the charitable purpose for which they donated, according to Felix et al. (2024). Also, voter participation and split-ticket voting decline, per Benton (2019) and other references discussed in "[[Information is a public good: Designing experiments to improve government]]". And the ultra-right does better, as noted in [[News from Germany 1900-1945 and implications for today]] and the section on "[[Information is a public good: Designing experiments to improve government#Previous research|Previous research]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]".<ref>Flößer (2024).</ref> The 0.13 percent of GDP savings estimated by Gao et al. (2018) is roughly $120 per human per year. With over 300 million humans in the U.S, that is roughly $40 billion nationwide. {| class="wikitable" |+ Table 1. Costs increase in news deserts |- ! Entity !! What !!Source |- | rowspan=4 |local government || costs +0.13% of GDP || Gao et al. (2018) |- | costs +0.05 - 0.11% of GDP || Baker and Murphy (2026 ) |- | pol. corruption charges +6.9% || Greenwood (2026) |- | pol. corruption charges +15% per member of INN || Usher and Kim-Leffingwell (2022) |- | local businesses || pollution & workplace accidents incr., innovation & econ growth decr. || Kim et al. (2021) |- | nonprofits || exec. compensation incr. || Felix et al. (2024) |- | rowspan=2 | elections | voter participation & split-ticket voting decl. || Benton (2019) |- | Ultra-right does better || Flößer (2024) |} === Government subsidies for news === John (1995) documented how in the first half of the nineteenth century the US had more independent newspaper publishers per million population than at any other time or place in human history.<ref>This is discussed in the 2025-06-08 [[Media concentration per Columbia History Professor Richard John|interview with him]], available on Wikiversity under [[:Category:Media reform to improve democracy]], accessed 2026-04-30.</ref> This encouraged literacy and limited political corruption, both of which helped [[The Great American Paradox|the early United States stay together and grow]] while contemporary [[w:New Spain|New Spain]] / [[w:Mexico|Mexico]], fractured, shrank, and stagnated economically. As documented with Figure 1 in the chapter below on [[/The impact of the media on political economy since the time of the Pharaohs/]], that growth catapulted the young United States into its current position of dominance in the international political economy, a position it has been losing since at least 1990 -- or since the Reagan Revolution began in 1981, according to the analysis in the chapter below on [[/Fox, the Great Depression, the Great Recession, and our future/]]. Other countries now have stronger democracies due in part to government subsidies for media in the range of 0.05 and 0.25 percent of GDP with a firewall that limits political interference in the content, according to Neff and Pickard (2024). Table 1 in "[[Information is a public good: Designing experiments to improve government]] compares media subsidies in various places with "other points of reference". McChesney and Nichols (2010, pp. 310-311, note 88) suggested that the relatively high rate of economic growth of the economy in the early US was due in part to postal subsidies under the US [[w:Postal Service Act|Postal Service Act]] of 1792.<ref>See also the Wikiversity article on "[[The Great American Paradox]]", accessed 2026-04-30.</ref> They estimated those subsidies at 0.21 percent of GDP. To improve the current political economy of the US, they recommended subsidies of 0.15 percent of GDP distributed to local news nonprofits on the basis of local elections.<ref>McChesney and Nichols (2021, 2022).</ref> The Wikipedia article on "[[Information is a public good: Designing experiments to improve government]]" documents how some jurisdictions can devote that much money to local news nonprofits by matching what they spend on accounting, advertising, and public relations.<ref>See the section on "[[Information is a public good: Designing experiments to improve government#Sampling units / experimental polities|Sampling units / experimental polities]]" in the Wikiversity article on "[[Information is a public good: Designing experiments to improve government]]", accessed 2026-04-30.</ref> Pickard (2023) describes three basic strategies for confronting concentrated commercial media power: (1) break them up, (2) regulate them, and (3) create non-commercial, public alternatives. A fourth possibility might be [[w:externality|a graduated tax on income and wealth]] in proportion to the threat that major corporations pose to democracy. One class of noncommercial alternatives that Pickard mentions is local multimedia / Public Media Centers (PMCs) with management split between local journalists and boards, e.g., selected at random from registered voters. A key here is to have the boards selected in a way that cannot be influenced by people with power, whether business or political elites. Picard recommends considering '''six discrete layers''' when discussing PMCs, each of which, he says, must be radically democratised: # funding, # governance, # ascertainment (to determine a community’s ''critical information needs''), # infrastructure (including universal broadband service), # algorithmic (e.g., not allowing companies like Google and Facebook to suppress indexing information the might challenge their hegemony of those markets, [[w:Deep web|treating them like pedophilia and the Islamic State]]), # engagement, involving local communities in making their own news and in communicating their own stories; this is paramount to building trust and the grassroots-level support that this new local journalistic model requires. All this needs to be managed in ways that provide substantive support to news deserts and underserved communities that have long been subjected to various kinds of informational redlining. This might be done by including the proposed PMCs within local libraries staffed by professional journalists, who provide training in media literacy in local schools for children and supervise students producing school newspapers. PMCs could host regular, e.g., monthly events, where local residents could share their concerns with journalist, who would use that input to help prioritize different issues for news coverage. Journalist could also coach local residents in how to research issues and collaborate with journalists in producing news reports that may be better researched and more relevant to local concerns than could be produced without such collaboration. Management of such PMCs might be split between journalists on staff and boards of, e.g., six members selected at random from voter registration rolls serving staggered terms of one year with a new member rotated in every 2 months. Another alternative that could be done in parallel with local PMCs calls for 200 journalists in each US Congressional district funded at $10 billion annually in 2022 dollars, which is just a little under 4 hundredths of one percent of GDP; if such allocations are expressed as fractions of a percent of GDP, they would grow naturally with the economy. (The nominal GDP for the US was roughly $26.1 trillion in 2022.<ref>Johnston and Williamson (2026).</ref> For 2026 it is estimated at $32.4 trillion.<ref>[[w:United States|United States]], accessed 2026-04-30.</ref>) A similar model is the [[w:BBC|BBC]]’s Local Democracy Reporting Service (LDRS), in which the BBC funds journalists to cover the work of local councils and other local public bodies, funded at £8 million per year, which is a little under 2 hundredths of a percent of the [[w:United Kingdom|UK]]'s GDP of £7.27 trillion.<ref>[[w:United Kingdom|United Kingdom]], accessed 2026-04-30.</ref> Pickard (2023) ended by saying, "Today we face a crossroads: technocracy and oligarchy from above or radical democracy and structural reform from below. ... [T]his is not just a journalism crisis: it is a democracy crisis." ==Table of Contents== *[[/Introduction/]] including an exercise, asking all to discuss perceptions of the settlement of ''[[w:Dominion Voting Systems v. Fox News Network|Dominion Voting Systems v. Fox News Network]]'' in a friendly supportive manner with humans with whom they may vehemently disagree, because the alternative could be killing humans over misunderstandings. ===Part I. The media and political economy=== # [[/The impact of the media on political economy since the time of the Pharaohs/]] describes how religious leaders in hierarchical societies prior to [[w:James VI and I|King James of the 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. During the reign of King James, pamphlets and newspapers began to compete with the church for helping commoners understand their roles in society. This produced the Industrial Revolution and modern democracies. Media consolidation since World War II slowed, then reversed this trend. # [[/Fox, the Great Depression, the Great Recession, and our future/]] describes the unprecedented performance of the US political economy during the presidency of Franklin Roosevelt (FDR), insisting that much of what FDR achieved can be replicated, giving a media system that supports honest discussion of the available evidence. # [[/Media consolidation, social media, and political polarization/]] (Combine from McChesney and Nichols discussing the [[w:Postal Service Act|US Postal Service Act]] of 1792 with [[Media concentration per Columbia History Professor Richard John]], the section on "[[v:Information is a public good: Designing experiments to improve government#Threats from social media|Threats from social media]]" in "[[Information is a public good: Designing experiments to improve government]], and the comments by [[v:Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen that, "the shortest path to a click is anger or hate."]]. ===Part II. The media and war=== # [[/Deterrence without threat/]]: The historical record is clear: Nations that have prepared for war often got war, not peace. This happens for at least two reasons: First, some leaders cannot resist the temptation to use force inappropriately, sometimes clandestinely provoking others to do things that are then denounced as "unprovoked"; sometimes the media environment pushes them to do such. Alternatively, potential adversaries may believe -- or claim -- that you are actually preparing a first strike, and they must move preemptively or lose their ability to retaliate adequately. We can avoid these possibilities with three supportive policies: [a] Legislation that ''prohibits'' projecting force beyond our own borders. [b] Civilian-based defense training in nonviolent noncooperation like what helped Denmark survive Nazi occupation with minimal damage. And [c] a media system that penalizes rather than encourages a bellicose foreign policy. # [[/Responding to a nuclear attack/]]: The ''worst'' response to a nuclear attack would be a nuclear response: The death toll from the 1945 [[w:Atomic bombings of Hiroshima and Nagasaki|Atomic bombings of Hiroshima and Nagasaki]] was estimated at between 150,000 and a quarter of a million. By contrast, simulations of nuclear wars of different magnitudes estimated that between 4 and 95 percent of humanity around the world would starve to death if they did not die of something else sooner. The range depends on whether the war was "limited", e.g., between India and Pakistan, and massive between the US and Russia. (Also add material from [[Nuclear weapons and effective defense]]). # [[/Threats from excessive government secrecy/]] (draft in [https://sanjosepeace.org/restrict-secrecy-more-than-data-collection/ "Restrict secrecy more than data collection"], adding material from [https://kkfi.org/program-episodes/does-us-government-secrecy-threaten-national-security/ Connelly (2023) ''The Declassification Engine: What History Reveals About America's Top Secrets''], [[Wikipedia:Moynihan Commission on Government Secrecy]] and [[1998 Embassy bombings and September 11]]. # [[/Shouting fire in a crowded theater/|Shouting ''fire'' in a crowded theater]]: Legal concerns about "[[w:Shouting fire in a crowded theater|Shouting ''fire'' in a crowded theater]]" date, at least in large part, from the [[w:Supreme Court of the United States|US Supreme Court]] decisions in ''[[w:Schenck v. United States|Schenck v. United States]]'' (1919) and ''[[w: Brandenburg v. Ohio| Brandenburg v. Ohio]]'' (1969). In ''Schenck'' the Court ruled that the government had a right to imprison Schenck and others, because their distribution of fliers encouraging draft resistance presented a [[w:clear and present danger|clear and present danger]] to the efficacy of ongoing military activities during [[w:World War I|World War I]], then in progress. The Court in ''Brandenburg'' held that the government cannot punish inflammatory speech ''unless that speech is "directed to inciting or producing imminent lawless action and is likely to incite or produce such action".'' Some could argue that many uses of military force by the US and Israel since 1948 have violated international law, encouraged by biases in the major US media "directed to inciting or producing imminent lawless action", though it may not be feasible to convince a court of that. Still, it might be useful to simulate such a case in a mock trial like the 1966 [[w:Russell Tribunal|Russell Tribunal]].<ref>Andersen (2006) provides such documentation for several such uses of force. Andersen (2026) and Johnson (2026) organize evidence supporting such claims for the current [[w:Gaza war|Gaza war]], which began with [[w:October 7 attacks|Palestinian attacks 2023-10-07]]. Might, e.g., Palestinians -- or at least Palestinian Americans -- be able to sue the [[w:Anti-Defamation League|Anti-Defamation League]] (ADL), the [[w:AIPAC|American Israel Public Affairs Committee]] (AIPAC), and all the major media outlets in the US for inciting genocide in the current [[w:Gaza war|Gaza war]]? That history includes routine suppression of coverage by the major media especially in the US of routine denial of equal protection of Israeli laws to non-Jews in Israel and under Israeli occupation, including suppression of Israeli violence against nonviolent protestors peaceably assembling and petitioning for a redress of grievances combined with over reporting of Palestinian violence and unquestioning coverage of fraudulent claims of Palestinian violence by Israel and supporters. The suppressions included underreporting of Palestinian nonviolence such as the [[w:2018–2019 Gaza border protests|(2018-2019) Great March of Return]], and suppression of the grievances inspiring such nonviolence such as indefinite detention without charges of thousands of Palestinians, including children, routine destruction of Palestinian property by settlers, confiscation of Palestinian property at gunpoint, closing [[w:Gaza Strip|Gaza]] to international trade, and maintaining Gaza on starvation rations. These routine biases in reporting have been encouraged by charges that more honest reporting would be "[[w:Antisemitism|antisemetic]], according to the ADL and AIPAC. This denial of coverage thereby encouraged Israel to increase the rate of such violations until the [[w:October 7 attacks|2023-10-07 attacks on Israel from Gaza]] unleashed Israeli "retaliations" way out of proportion to the alleged provocation. Sucharov (2022) reported that 69% of American Jews opposed privileging Jews over non-Jews in Israel. Their support of Israel in the current Gaza war is consistent with the media biases documented by Johnson (2026) and others including Andersen (2006, 2026). Regarding whether Israel could achieve anything positive from this war, Samuelson (2025) is skeptical. He summarized quantitative analyses of 60 previous insurgencies. The results including the observation that it is exceedingly difficult to defeat an insurgency without responding to the grievances that support it without force ratios far beyond Israel's resources.</ref> ===Part III. Climate, immigrants, education, public health, and criminal justice=== # [[/Global warming/]] [Summarize research especially on conflicts of interest of major media in honestly reporting on this issue and the research on global warming itself and activities of groups concerned about this issue. Decompose into global population times CO2 equivalents per human.] # [[/Immigrants/]] [Summarize research documenting that [[w:Sanctuary city|sanctuary cities tend to have higher median incomes and no more crime than non-sanctuary jurisdictions]], and some studies report less crime. Moreover economists have documented that immigrants tend to be more entrepreneurial, overrepresented in patent applications, and generally increasing the rate of economic growth. See, e.g., Aghion et al. (2022) ''The power of creative destruction''; Aghion shared the 2025 Nobel Memorial Prize in Economics with two others.] # [[/Education/]] (draft in [[Invest in children]].) # [[/Public health/]] [Draft in [[UN public health data]] to be revised to be consistent with Bezruchka (2023, 2025).] # [[/Substance abuse and addictive behavior/]] (Research cited in "[[Wikipedia:War on drugs]]" insists that the US and the world would have fewer problems with substance abuse and addiction problems with 100 percent public funding for treatment programs and complete decriminalization of possession and use of retail quantities of addictive substances. We would also likely have fewer problems with immigrants, as that would make it harder for the US to intervene in the internal affairs of foreign countries funded off the books, as exposed in the [[w:Iran–Contra affair|Iran–Contra affair]]. See [[w:Alfred W. McCoy|Alfred W. McCoy]] (1972; 3rd ed. 2003) ''The politics of heroin''. Journalist [[w:Gary Webb|Gary Webb]] published similar claims that the US intervention in Nicaragua in the Iran-Contra Affair was funded in part by drug money. The story was officially retracted by the ''San José Mercury News'', and Webb allegedly killed himself with 2 bullets to the head. # [[/Criminal justice/]] (The section on "[[w:United States incarceration rate#Editorial policies of major media|Editorial policies of major media]]" in "[[Wikipedia:United States incarceration rate]]" cites research claiming that within the range range of experience in the US political economy since 1925, the incarceration rate is uncorrelated with crime: It's a function of the public's perception of crime, and that's a function of the media. That suggest that the US would be safer and more prosperous if incarceration policies were driving more by research than by editorial policies of the media. For example, there is also research that says that incarcerees who receive visits are less likely to recidivate, but that evidence is overlooked when convicts are incarcerated substantial distance from their family and friends and when the cost of phone services is substantially higher for incarcerees than among the general pubic. Also, it's known that better educated incarcerees are less likely to recidivate, but it's difficult and maybe impossible for many incarcerees to obtain education in prison.) # [[/Empower women and girls/]] [Cite research claiming that a primary restraint on population growth is empowering women and girls. Empowering women and girls is not just a matter of equity: It is also a means to reduce the threats of global warming, of increasing exposure to animal diseases and other problems that come with unrestrained population growth.] === Continuation === * [[/The evolving media literacy movement/]] to invite others to keep this book current with the evolving understanding of media literacy, how to encourage and promote it and the benefits of doing so. ==See also== * [[Wikibooks:Antiracist Activism for Teachers and Students/Points to Consider for Teaching Anti-racism/Media Literacy In Schools]] ==Notes== {{reflist}} ==Bibliography== * <!--Daron Acemoğlu and Simon Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Robin Andersen (2006) A century of media, a century of war-->{{cite Q|Q138795568}} * <!--Robin Andersen (2026-06-02) The Complicit Lens: US Media Coverage of Israel’s Genocide in Gaza-->{{cite Q|Q138796307}} * <!-- Matthew Baker and Dermot Murphy (2026-06-10) "Local News Shortage Leads to $1.1 Billion in Extra Borrowing Costs for Local Governments and Taxpayers"-->{{cite Q|Q141620048}} * <!--Perry Bacon Jr. (2022-10-17) "America Should Spend Billions to Revive Local News"-->{{cite Q|Q139594786}} * <!-- Joshua Benton (9 April 2019). "When local newspapers shrink, fewer people bother to run for mayor". Nieman Foundation for Journalism -->{{cite Q|Q63127216}} * <!--Stephen Bezruchka (2023) Inequality Kills Us All-->{{cite Q|Q136047815}} * <!--Stephen Bezruchka (2025) ''Born Sick in the USA''-->{{cite Q|Q138749292}} * <!--Renée DiResta (2024) Invisible Rulers: The People Who Turn Lies into Reality-->{{cite Q|Q135530439}} * <!--Robert Felix, Joshua A. Khavis, and Mikhail Pevzner (2024) "The effects of local newspaper closures on nonprofits’ executive compensation"-->{{cite Q|Q132730972}} * <!--Maxim Flößer (2024-03-06) "Keine Lokalzeitung -- mehr AfD", Kontext-->{{cite Q|Q125287792}} * <!--Pengjie Gao, Chang Lee, and Dermot Murphy (2018) "Financing Dies in Darkness? The Impact of Newspaper Closures on Public Finance"-->{{cite Q|Q55670016}} * <!--Spencer Graves (2024) "Wikipedia: The most democratic force on earth-->{{cite Q|Q137796922}} * <!--Spencer Graves and Bryan Bailey (2025) "We have to talk", blog at PeaceWorksKC.org-->{{cite Q|Q136126262}} * <!--Brad N. Greenwood (2026-06-09) "When Local Newspapers Die, Corruption Festers"-->{{cite Q|Q141620523}} * [[d:Q138038060|Dan Hind and Spencer Graves (2025) "Media Reform Coalition challenges anti-democratic media bias in the UK" on Wikiversity]]. * <!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse-->{{cite Q|Q54641943}} * <!--Adam H. Johnson (2026-04-21) How to Sell a Genocide: The Media's Complicity in the Destruction of Gaza-->{{cite Q|Q140073447}} * <!--Louis Johnston and Samuel H. Williamson, "What Was the U.S. GDP Then?" MeasuringWorth, 2026-->{{cite Q|Q56881105}} * <!-- Min Kim, Derrald Stice, Han Stice, and Roger M. White (2021) "Stop the presses! Or wait, we might need them: Firm responses to local newspaper closures and layoffs"-->{{cite Q|Q132459373}} * <!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) -->{{cite Q|Q104888067}}. * <!-- Robert W. McChesney; John Nichols (2021). "The Local Journalism Initiative: a proposal to protect and extend democracy". Columbia Journalism Review, 30 November 2021 -->{{cite Q|Q109978060}} * <!-- Robert W. McChesney; John Nichols (2022), To Protect and Extend Democracy, Recreate Local News Media (PDF), FreePress.net (updated 25 January 2022) -->{{cite Q|Q109978337|access-date=2024-06-23}} * <!--Alfred W. McCoy (2003-05-01) The politics of heroin : CIA complicity in the global drug trade : Afghanistan, Southeast Asia, Central America, Colombia-->{{cite Q|Q141317509}} * <!--Joel Mokyr (2017) A Culture of Growth: The Origins of the Modern Economy (Graz Schumpeter Lectures; Princeton University Press)-->{{cite Q|Q136118590}} * <!--Neff and Pickard (2024) "Funding Democracy: Public Media and Democratic Health in 33 Countries"-->{{cite Q|Q131468289}} * [[d:Q131398359|Victor Pickard (2020) ''Democracy without journalism? : confronting the misinformation society'' (Oxford U. Pr.)]]. * <!-- Victor Pickard (2023-05-12) "Another Media System is Possible: Ripping Open the Overton Window, from Platforms to Public Broadcasting"-->{{cite Q|Q131398460}} * <!--Doug Samuelson (2025) Assessing Israel’s Approach in Gaza-->{{cite Q|Q138843324}} * [[d:Q138037937|Dean Starkman and Spencer Graves (2025) "Dean Starkman and the watchdog that didn't bark anglais" on Wikiversity]]. * <!--Mira Sucharov (2022) Do American Jews Really Know What 'Zionist' Means?-->{{cite Q|Q125903777}} * [[d:Q134715465|Nikki Usher and Sanghoon Kim-Leffingwell (2022) "How Loud Does the Watchdog Bark? A Reconsideration of Local Journalism, News Non-profits, and Political Corruption", ''SSRN Electronic Journal'']]. * [[d:Q61013892|Horacio Verbitsky (1997) ''Un mundo sin periodistas'' (in Spanish: A world without journalists; Editorial Sudamericana)]]. [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Psychology]] [[Category:Sociology]] [[Category:Education]] [[Category:Economics]] [[Category:Media Literacy and You]] [[Category:Freedom and abundance]] <!-- https://en.wikiversity.org/wiki/Category_Review --> tuy048wn1mb553baewsptssg3hgya27 0 330941 2835504 2835157 2026-10-02T12:43:18Z U3269672 3105208 /* Akrasia */ 2835504 wikitext text/x-wiki {{title|Akrasia:<br>Why do people act against their better judgement?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}}'''Consider this scenario''' Jordan signed up for a gym membership at the start of the year, determined to improve his physical and mental health. He bought new workout clothes, saved a list of routines and even told his friends he was committed to exercising. But when his alarm goes off at 6:30am, he hits snooze, telling himself he will simply go after work instead. By 5:00pm, he's too tired to go. The idea of lifting weights is too emotionally heavy and stress inducing for him, so Jordan chooses to stay home and watch a TV show. Each time Jordan's alarm goes off, he ignores it, even though he knows that going to the gym is a healthy option and helps him gain confidence although the short-term comfort of staying at home outweighs those feelings. Jordan isn't lazy, he's experiencing akrasia, a predictable motivational and emotional conflict between long term goals and short-term impulses. {{RoundBoxBottom}} [[File:Woman Stressed At Work Illustration.jpg|thumb|Figure 1. A women experiencing stressful situation with conflicting priorities. ]] This Chapter explains and explores why people act against their better judgment. It describes this through psychological explanations of self-control and regulation, motivational deficits, and emotional avoidance. This phenomenon has been named Akrasia. '''The problem''': Akrasia is something most people will experience in their life. The problem with akrasia is that it can affect many areas of an individual's life, such as academics, health choices, financial decisions, and relationships. This is because many people intend to make choices that align with their values and goals, yet often make choices that move them further away from them. This gap between intention and action can lead to issues with wellbeing and a sense of success in life. Understanding akrasia is important because it undermines goal achievement, wellbeing, and personal goals. It reflects the tension between reflective, value-based intentions and impulsive actions. Understanding the psychological science behind this concept matters because it gives you tools to understand the conflict by examining self-regulation and cognitive bias. This will also give you techniques to help reduce akrasia so you can reach the goals you have set.{{RoundBoxTop|theme=5}}'''Focus questions''' [[File:Crystal Clear app ktip.svg|left|20px|]] * What is akrasia? * What psychological processes contribute to akrasia and the conflict between long term and short term impulses? * How do motivation and emotion interact to influence why people act against their better judgement ? * What evidence based strategies can help people reduce akrasia and act more consistently in line with their goals ? {{RoundBoxBottom}} == Akrasia == ==== What is Akrasia ==== The term Akrasia refers to the psychological phenomenon of acting against one's better judgment. Psychological research behind this phenomenon explains it as a product of predictable conflicts within the human self-regulation system. Rather than a moral weakness, akrasia reflects the tension between the reflective, long-term intentions and immediate, emotionally driven short-term impulses. Studies done on self-regulation highlight how failures in goal monitoring, susceptibility to short-term rewards and reduced sensitivity to long-term consequences make an individual more likely to behave in ways that contradict their own goals. Motivation research further describes that self-indulgence impulses- such as seeking comfort or avoiding discomfort can overpower reflective decision-making, producing patterns of avoidance. In sum Akrasia involves cognitive, emotional, and motivational processes and together, they can describe how and why akrasia occurs. [[File:Aristotle Altemps Inv8575.jpg|thumb|Figure 2. An image of Aristotle ]] ==== Historical background: where did the term Akrasia come from? ==== Historically, Akrasia has been recognised for thousands of years. The concept originates in ancient Greek philosophy, particularly in the work of Aristotle, who used the term akrasia to describe "weakness of will". Aristotle argued that people sometimes fail to act in accordance with their rational understanding because emotions, desires, or impulses temporarily overpower reason. In his view, akrasia occurs when the rational part of the mind identifies the best course of action. However, the nonrational part disrupts this judgment and pulls behaviour toward short-term gratification. This failure is not due to ignorance but to momentary inability to manage or correctly interpret one's emotional and motivational states. Earlier, Socrates debated whether akrasia was even possible, suggesting that if someone acted wrongly, it must be because they misunderstood what was truly good. Aristotle disagreed, emphasising that people can genuinely know the right action yet still fail to follow it. He suggested that individuals often misjudge the force of their impulses, treating them as rational justifications in the moment. This historical debate helped lay the foundation for modern psychological explanations. (Bobonich & Destree, 2007). == Akrasia as a motivational and emotional conflict == '''Focus question''': what psychological processes contribute to akrasia and the conflict between long term and short term impulses? Akasia is easier to understand when framed as a psychological conflict between internal systems that operate under different rules, time scales, and motivational priorities (Frederick, Loewenstein, & O’Donoghue, 2002). From a psychological perspective, long-term goals are maintained by reflective, value-based processes that rely on executive functioning. This includes future planning and a coherent sense of identity- the part of the mind that says "finishing my degree matters for my future". These goals are cognitively effortful, meaning they require working memory, sustained attention, and the ability to tolerate discomfort for the reward of a future outcome. In contrast, short-term impulses arise from fast, affect-driven systems that prioritise immediate emotional states. In basic terms, when someone feels tired, stressed or bored, their short-term system generates urges like "I just want to relax now". These impulses are not irrational or because of laziness they are grounded in the brain's reward circuitry, which is highly sensitive to immediate relief and comfort (Sirois & Pychyl, 2013). Psychology research shows that akrasia occurs most often when tasks evoke negative affect, such as states of anxiety, frustration, or uncertainty. Under these conditions, people tend to prioritise short-term affective rewards, such as relief from discomfort, even when they consciously prefer the long-term outcomes. This means that akrasia can serve as a mood repair strategy, where avoidance temporarily improves emotional states at the cost of long-term goals (Steel, P. (2007). Akasia can be seen as both a positive and negative cycle because while the short term relief feels nice the ramifications of the long terms getting harder to reach can feel worse. Dual-process models explain this dynamic by distinguishing between two interacting systems: a fast, automatic, emotionally reactive system called system 1 and a slower, reflective, future-oriented system called system 2. Akrasia is when system 1 dominates behaviour in the moment, overpowering the deliberate reasoning of system 2.  Factors such as cognitive load, stress, and emotional discomfort weaken an individual's reflective control, making impulsive choices more likely.  All this to say, akrasia results from how human motivational and emotional systems compete for behavioural control (Hofmann, Friese, & Strack, 2009).  == Self‑regulation and self‑control == '''Failures in self‑regulation contribute to akrasia''': Self-regulation is a multifaceted process that describes an individual's ability to manage emotions, thoughts, and behaviours in ways that yield positive results. Failure occurs when an individual cannot coordinate behaviours, emotions, or attention to achieve a long-term goal (Catanese, 2024). While researchers cannot yet identify a single cause of self-regulation failures, we can identify patterns that produce it (Baumeister & Heatherton, 1996). Self-regulation failure can cause significant harm by leading individuals to act in ways that may harm them or others financially, physically, or emotionally. Self–regulation failure can be broken down into different conceptualisations of how it occurs; one is the three-part feedback loop proposed by Carver and Scheier (1981).   The first part describes standards, the ideals and goals; without proper standards, self-regulation is hindered. The second part is monitoring; this is where individuals must account for their actions in relation to their standards, meaning that if someone is not holding themselves accountable to meet their standards, failure often starts here.  The third part of operating is when monitoring falls short of the standards, and the process of changing the goal or state begins.  Failure occurs because, while a person can set a goal and monitor effectively, they may still fail to achieve it (Baumeister & Heatherton, 1996). Bella (2023) describes another way self-regulation failure can occur, with similar underlying factors to Carver's and Scheier's (1981).  She argues that akrasia is best understood as a conflict between internal systems. Specifically, the reflective system represents long-term values and goals, while the impulsive system responds to immediate emotional and motivational states. When these systems compete, behaviour becomes inconsistent, fragmented, and vulnerable to short-term impulses. Failure indicates that the individual's ability to manage impulses and emotions and maintain goal-directed behaviour is compromised. This vulnerability is not a character flaw, but a predictable psychological pattern that arises when regulatory processes are overloaded, undermined, and not integrated effectively. More precisely, akrasia occurs when the impulsive system becomes more dominant because short-term impulses feel more compelling, and long-term goals feel more distant and effortful. This is what makes a person act against their better judgment, even when they fully understand the potential long-term consequences. (Bella, 2023) Understanding self-regulation failure would help identify solutions to acts of akrasia and other motivation conflicts. Understanding allows an individual to alter their response to a stimulus. '''Discuss ego depletion''' Another concept that helps describe why akrasia might occur is ego depletion.  Ego depletion is the idea that self-control operates like a limited resource. When people expend self-control, they regulate their emotions, resist temptations, make decisions, and force themselves to finish difficult tasks. Once this limited resource of self-control is depleted, their ability to persist, inhibit impulses, or act according to long-term goals becomes significantly impaired.   Baumeister, Bratislavsky, and Tice (1998) proposed that self-control relies on a single, general resource. This means any act of self-control consumes this resource, leaving a person temporarily less capable of exerting further control.  When individuals become depleted, they are more vulnerable to short-term impulses, making akrasia more likely. This is possible even when the long-term goal is important; they lack the regulatory strength to act on it. More recent work by Inzlicht and Schmeichel (2012) challenges the idea of self-control as a resource, arguing instead that ego depletion reflects shifts in motivation, attention, and affect. After exerting self-control, people become less motivated to exert effort and less attentive to long-term goals. This still supports the idea that akrasia is a regulatory system that shifts towards seeking short-term rewards over long-term intentions.  These processes make akrasia more likely, as individuals who are depleted struggle to inhibit impulses or tolerate discomfort. In everyday life, ego depletion can be seen when a student intends to study after a long day but instead chooses to relax because their regulatory capacity is drained.  In these moments, the long-term goal remains valued, but the ability to fulfil it is compromised.{{Robelbox|theme=5|width=60%|title=1.1 - Module overview|iconwidth=47px|icon=Nuvola_apps_korganizer.svg}}<div style="{{Robelbox/pad}}"> <quiz display=simple> { Which subject best explains how failures in self regulation contribute to akrasia? } |type="()"} - Ego Depletion - Temporal discounting + Internal system conflict </quiz> </div> {{Robelbox-close}} == Motivational theories == [[File:Self-Determination-Theory-Visual 1.png|thumb|Figure 3: Self-Determination Theory visual diagram ]] ===== Self‑determination theory: ===== Self-determination theory explains innate psychological needs in relation to human motivation (Ryan & Deci, 2000). It was developed by Edward Deci and Richard Ryan (1985). The theory describes how humans have an innate predisposition for psychological growth and development, fueled by three basic psychological needs. Needs are defined as inherent and motivating (Ryan & Deci, 2017). Once fulfilled, these needs contribute to a person's health and well-being. The first need identified is autonomy. It is when one's behaviour is self-directed and congruent with their personal interests and values. This need is satisfied when an individual's actions fulfil their desires and feel authentic to them (Deci & Ryan, 2015).   The second is competence: the urge to feel capable and develop a mastery of performing tasks.  To satisfy this, an individual can challenge their skills and abilities (Vansteenkiste et al., 2020). Lastly, relatedness is the feeling of connection and belonging. It is fulfilled when an individual feels capable of caring for others and being cared for. When an individual's needs are not satisfied, however, this can result in illness or discontent (Deci & Ryan, 2000).  STD proposes that not all motivation is equal, meaning that the focus should be on how and why a person is motivated. To understand why akrasia occurs, it is crucial to understand the difference between autonomous and controlled motivation. Autonomous motivation includes two forms: intrinsic motivation, which refers to engaging in an act because it is personally meaningful and interesting. Integrated regulation refers to acting because it’s aligned with one’s values and sense of self (Deci, 1972).  Individuals must feel a sense of choice, volition, and psychological freedom, rather than pressure to perform.  When these needs are met, individuals show satisfaction with the three needs.  Akrasia becomes less likely when motivation is autonomous. When people genuinely value a goal, they are more likely to tolerate discomfort, delay gratification, and persist even when it is difficult. This is because the action feels internally meaningful, individuals feel capable of reaching their end goal, and they feel supported in their action, so short-term impulses have less power.  An individual may still act autonomously, even in accordance with external expectations, if they believe their behaviour remains consistent with their values and is of their own volition (Ryan & Deci 2017). Controlled motivation describes the feelings of pressure and tension. It occurs when behaviour is driven not by personal values or genuine interest but by external demands, internal pressure or avoidance of guilt and shame. This is the opposite of autonomous motivation and often causes akrasia. Similar to autonomous motivation, it breaks down into two main forms. External regulation: acting to obtain a reward or avoid punishment.  Introjected regulation, acting to avoid guilt, anxiety, or maintain self-worth.  In both cases, behaviour tends not to be fully self-endorsed.  Controlled motivation increases akrasia; it makes tasks feel aversive, even when the individual knows the long-term goal is important, because this creates emotional friction and associates tasks with stress, guilt, or obligation.  When an individual feels controlled, they are more likely to avoid the task, procrastinate, or act against their better judgment (Deci et al., 2012).{{RoundBoxTop|theme=5}} ''' Example scenario''' Annie really wants to get her psychology degree, but she finds studying difficult however she continues to engage and attend class to achieve her goal. She is pushing through a desire to disengage, procrastinate, and stop in order to feel a sense of achievement that she did herself. {{RoundBoxBottom}} *expectancy‑value theory: actions based an appraisal of successes (reference: Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. ''Contemporary Educational Psychology'', ''25''(1), 68–81.https://doi.org/10.1006/ceps.1999.1015) *Show how motivation quality affects the likelihood of acting in line with long‑term goal * emotional avoidance: tasks that evoke frustration are avoided this is a form of regulation ( reference : De Castella, K., Platow, M. J., Tamir, M., & Gross, J. J. (2018). Beliefs about emotion: implications for avoidance-based emotion regulation and psychological health. ''Cognition and Emotion'', ''32''(4), 773–795. https://doi.org/10.1080/02699931.2017.1353485) <quiz display="simple"> {SDT is primarly concerned with ? :} - How rewards increase performance + How basic psychological needs influence motivation - How motivation shapes decision making {SDT views motivation as ?:} - A simple intrinsic/extinsic split - A set of unrelated catagories + A continuum from controlled to autonomous </quiz> == Reduction Techniques == We have now discussed how and why akrasia occurs. Let's move onto techniques you can use in your day-to-day life to reduce akrasia.   Reducing cognitive load though targeted behavioral and environmental techniques is an effective method to decrease akrasia because these strategies simplify task demands and support sustained self-regulation.  The techniques include task chunking, breaking up complex activates into smaller more cognitively manageable sections, lowering the initiation barrier, and reducing the likelihood of avoidance (sweller, 1988). Externalizing memory, this is creating an organized method of reminder though planners and checklists in order to off load working memory demands, allowing the individual to maintain goal relevant information without experiencing cognitive strain, thereby improving follow through on intentions ( sweller et al, 2011). Environmental simplification is the process of reducing visual clutter and minimizing digital distractions. This will decrease extraneous cognitive load and help preserve attentional resources for goal-directed behavior. Similarly, decision reduction strategies which is the process of cutting down options such making an established routine or making default choices, this can limit the cognitive effort required for repeated decision making, protecting self-regulatory capacities from depletion (Baumeister & Heatherton, 1996). Together, these techniques reduce unnecessary cognitive demands and help create conditions in which intentions are more likely to become actions.   Another way to reduce akrasia is through enhancing autonomous motivation. This is to shape the environment so that you individuals can experience volition, psychological freedom and personal endorsement.  Reeve & Jang (2006) discuss this idea, of autonomy support, stating that increases in autonomous motivation helps an individual feel their behavior is self-initiated rather than externally pressured. They demonstrated this by studying what teachers say and do during learning activities that helped increase student engagement and intrinsic motivation.  one of the first things they recognized what that acknowledging that learners' thoughts, feelings and frustrations show respect for their internal experience. This works because it allows the individual to feel their task is not being controlled by others. A second finding shows that allowing oneself to explore, reflect and attempt the task without interruption this supports autonomy and competence. A third finding is that the feedback you give yourself should be informal and not controlling meaning changing the mindset to find alternative ways to think about a task, looking at an issue with a new more light perspective rather than a negative one such as " this should be done this way" to "how many ways are there to reach this goal, task, problem". This helps enhance competence and strengthen motivation.  <quiz display="simple"> {Which technique would you use? :} - Reduction of cognitive load - Autonomous motivation - Goal Setting </quiz> == The Understanding of Psychological Science Of Akarsia and Support == '''Focus question''': what strategies are there to reduce to act of akrasia and act more consciously in line with their goals? Empirical fining both support and refine theoretical models of akrasia by showing that it arises from multiple interaction processes, rather than simple tendency to put of work, reach on self-regulation failure(Bella, 2024), ego depletion (Baumeister et al., 1998; inzlicht & schmeichel, 2012) and self-determination theory (Deci & Ryan, 200) collectively demonstrate that akrasia reflects conflicts between long term, value based intentions and short term, affect driven impulses, shaped by motivational quality, emotional avoidance and cognitive load. These findings challenge overly simplistic models that treat akrasia as simply procrastination or lack of willpower, and instead support integrative frameworks in which internal systems, basic psychological needs and situational demands jointly determine whether people act in line with their better judgement. This integrated knowledge has important implications for improving motivational and emotional lives. Interventions should not only focus on trying harder, but also support autonomous motivation and emotional regulation while reducing unnecessary control and guilt. By targeting these mechanisms, individuals are more likely to experience motivation that is self-endorsed, resilient to discomfort, and aligned with long-term goals (Sirois & Pychyl, 2013).    * {{RoundBoxTop|theme=5}} ADD scenario that follows up on jordans problem with a solutions based on principles discussed {{RoundBoxBottom}} ==Conclusion== * summarise the most important points in 3–5 sentences. * Align with subtitle and focus questions. * Highlight implications for personal growth and psychological wellbeing. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[Motivation and emotion/Book/2011/Self-determination theory|Self-determination theory]] (Book chapter, 2011) * [[wikipedia:Goal_setting|Goal setting]] (Wikipedia) * [[wikipedia:Motivation|Motivation]] (Wikipedia) ==References== {{Hanging indent|1= Bobonich C., & Destrée, P. (2007). Akrasia in Greek Philosophy: From Socrates to Plotinus. BRILL .106, 1-11. https://books.google.com.au/books?id=so097j_jPBsC&dq=akrasia+historical+background&lr=&source=gbs_navlinks_s Bella, A. F. (2023). Psychological underpinnings of akrasia: A new integrative framework based on self-regulation vulnerabilities and failures. New Ideas in Psychology, 70, Article 101027. https://doi.org/10.1016/j.newideapsych.2023.101027) Baumeister, R. F., et al. (1998). Ego depletion: Is the active self a limited resource? Journal of Personality and Social Psychology. DOI: https://doi.org/10.1037/0022-3514.74.5.1252 (doi.org in Bing) Deci, E. L., & Ryan, R. M. (2000). The “What” and “Why” of Goal Pursuits: Human Needs and the Self-Determination of Behavior. Psychological Inquiry, 11(4), 227–268. https://doi.org/10.1207/S15327965PLI1104-01 De Castella, K., Platow, M. J., Tamir, M., & Gross, J. J. (2018). Beliefs about emotion: implications for avoidance-based emotion regulation and psychological health. Cognition and Emotion, 32(4), 773–795. https://doi.org/10.1080/02699931.2017.1353485 Frederick, S., Loewenstein, G., & O’Donoghue, T. (2002). Time discounting and time preference: A critical review. Journal of Economic Literature, 40(2), 351–401. https://doi.org/10.1257/002205102320161311 Hofmann, W., Friese, M., & Strack, F. (2009). Impulse and self-control from a dual-systems perspective. Perspectives on Psychological Science, 4(2), 162–176. https://doi.org/10.1111/j.1745-6924.2009.01116.x Inzlicht, M., & Schmeichel, B. J. (2012). What Is Ego Depletion? Toward a Mechanistic Revision of the Resource Model of Self-Control. Perspectives on Psychological Science, 7(5), 450–463. https://doi.org/10.1177/1745691612454134 Locke, E. A., & Latham, G. P. (2002). Building a practically useful theory of goal setting and task motivation: A 35-year odyssey. The American Psychologist, 57(9), 705–717. https://doi.org/10.1037/0003-066X.57.9.705) Reeve, J., & Jang, H. (2006). What Teachers Say and Do to Support Students’ Autonomy During a Learning Activity. Journal of Educational Psychology, 98(1), 209–218. https://doi.org/10.1037/0022-0663.98.1.20 Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285. https://doi.org/10.1207/s15516709cog1202_4 Sirois, F. M., & Pychyl, T. A. (2013). Procrastination and the priority of short-term mood regulation: Consequences for future self. Personality and Individual Differences, 55(6), 653–658. https://doi.org/10.1016/j.paid.2013.06.011 Steel, P. (2007). The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure. Psychological Bulletin, 133(1), 65–94. https://doi.org/10.1037/0033-2909.133.1.65 Wigfield, A., & Eccles, J. S. (2000). Expectancy–Value Theory of Achievement Motivation. Contemporary Educational Psychology, 25(1), 68–81. https://doi.org/10.1006/ceps.1999.1015}} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== * [https://pmc.ncbi.nlm.nih.gov/articles/PMC3062191/ Cognitive neuroscience of self regulation failure] (National library of medicine) {{ic|This should be in references and cited}} * [https://www.youtube.com/watch?v=jHGFlyHnhFQ Self control and Akrasia] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-control]] m15n88c988jw0cb02v43n42decjufnh 0 331046 2835513 2835330 2026-10-02T13:43:58Z Flickstar888 3106813 /* How self-concept can undermine motivation */ 2835513 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world. This chapter explores how the understanding of oneself influences what people are motivated to do and examines how motivated behaviour can shape self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth. {{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related, but unique to many of the popular “self-” constructs like [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me”, the material, social, and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of other peoples' responses and evaluations. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become), and ought self (who one believes others want them to be; see Table 2). Ideal and ought selves set standards for one to strive toward, laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focuses positively correlated with creativity and an innovative cognitive style, whereas prevention focuses negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self-identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges. These states modify goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person, and how they subsequently behave, are shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz - Test yourself}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How people feel about themselves + Standards people feel they must meet to have value - How people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, because of how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, result in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure cannot be self-attributed (Török & Szabó, 2018). A handicap acts as a plausible external explanation for an outcome other than the self. This means that a handicapper is willing to increase the probability of failure to protect their self-concept (Coudevylle et al., 2020). Therefore, self-concept can impact motivation by hindering goal pursuit. {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical, multi-faceted construct, and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). This means interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concepts, and growth-oriented interpretations of failure to improve motivation and wellbeing.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence, and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation (Table 4). These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined people's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by exercise motivation. This result conflicts with the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable, and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change. Identity-based motivation theory explains how active identities change the meaning of difficulty. Whereas self-worth contingencies can undermine motivation. Together, these theories detail that self-concept influences goal direction and engagement through social, emotional and cognitive processes. These theories provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] dhoq9wijoovtfwvn44ka6aunlcfm1ii 2835514 2835513 2026-10-02T13:47:49Z Flickstar888 3106813 Added 'Quiz' to box 2835514 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world. This chapter explores how the understanding of oneself influences what people are motivated to do and examines how motivated behaviour can shape self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth. {{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related, but unique to many of the popular “self-” constructs like [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me”, the material, social, and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of other peoples' responses and evaluations. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become), and ought self (who one believes others want them to be; see Table 2). Ideal and ought selves set standards for one to strive toward, laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focuses positively correlated with creativity and an innovative cognitive style, whereas prevention focuses negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self-identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges. These states modify goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person, and how they subsequently behave, are shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> Quiz '''What are self-worth contingencies?''' |type="()"} - How people feel about themselves + Standards people feel they must meet to have value - How people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, because of how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, result in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure cannot be self-attributed (Török & Szabó, 2018). A handicap acts as a plausible external explanation for an outcome other than the self. This means that a handicapper is willing to increase the probability of failure to protect their self-concept (Coudevylle et al., 2020). Therefore, self-concept can impact motivation by hindering goal pursuit. {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical, multi-faceted construct, and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). This means interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concepts, and growth-oriented interpretations of failure to improve motivation and wellbeing.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence, and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation (Table 4). These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined people's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by exercise motivation. This result conflicts with the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable, and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change. Identity-based motivation theory explains how active identities change the meaning of difficulty. Whereas self-worth contingencies can undermine motivation. Together, these theories detail that self-concept influences goal direction and engagement through social, emotional and cognitive processes. These theories provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] l09sl1wcmetxrei9ow08nx97ftbeioz 2835639 2835514 2026-10-03T06:54:34Z U3253363 3106362 /* How self-concept can undermine motivation */ trying to move the added heading of 'quiz' 2835639 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world. This chapter explores how the understanding of oneself influences what people are motivated to do and examines how motivated behaviour can shape self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth. {{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related, but unique to many of the popular “self-” constructs like [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me”, the material, social, and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of other peoples' responses and evaluations. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become), and ought self (who one believes others want them to be; see Table 2). Ideal and ought selves set standards for one to strive toward, laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focuses positively correlated with creativity and an innovative cognitive style, whereas prevention focuses negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self-identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges. These states modify goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person, and how they subsequently behave, are shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11}} <quiz display=simple> Quiz '''What are self-worth contingencies?''' |type="()"} - How people feel about themselves + Standards people feel they must meet to have value - How people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, because of how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, result in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure cannot be self-attributed (Török & Szabó, 2018). A handicap acts as a plausible external explanation for an outcome other than the self. This means that a handicapper is willing to increase the probability of failure to protect their self-concept (Coudevylle et al., 2020). Therefore, self-concept can impact motivation by hindering goal pursuit. {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical, multi-faceted construct, and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). This means interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concepts, and growth-oriented interpretations of failure to improve motivation and wellbeing.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence, and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation (Table 4). These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined people's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by exercise motivation. This result conflicts with the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable, and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change. Identity-based motivation theory explains how active identities change the meaning of difficulty. Whereas self-worth contingencies can undermine motivation. Together, these theories detail that self-concept influences goal direction and engagement through social, emotional and cognitive processes. These theories provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. ''Instructional Science, 52'', 89–107. https://doi.org/10.1007/s11251-023-09642-8 Cooley, C. H. (1902). The looking-glass self. ''The production of reality: Essays and readings on social interaction, 6''(1902), 126–28. Coudevylle, G. R., Boulley-Escriva, G., Finez, L., Eugène, K., & Robin, N. (2020). An experimental investigation of claimed self-handicapping strategies across motivational climates based on achievement goal and self-determination theories. ''Educational Psychology, 40''(8), 1002–1021. https://doi.org/10.1080/01443410.2020.1746237 Garn, A., & Shen, B. (2015). Physical self-concept and basic psychological needs in exercise: Are there reciprocal effects? ''International Journal of Sport and Exercise Psychology, 13''(2), 169–181. https://doi.org/10.1080/1612197X.2014.940994 Higgins, E. T. (1987). Self-discrepancy: A theory relating self and affect. ''Psychological Review, 94''(3), 319. Higgins, E. T. (1989). Self-discrepancy theory: What patterns of self-beliefs cause people to suffer?. In ''Advances in experimental social psychology'' (Vol. 22, pp. 93–136). Academic Press. Higgins, E. T., Shah, J., & Friedman, R. (1997). Emotional responses to goal attainment: Strength of regulatory focus as moderator. ''Journal of Personality and Social Psychology, 72''(3), 515–525. https://doi.org/10.1037/0022-3514.72.3.515 Hübner, N., Wagner, W., Zitzmann, S., & Nagengast, B. (2023). How strong is the evidence for a causal reciprocal effect? Contrasting traditional and new methods to investigate the reciprocal effects model of self-concept and achievement. ''Educational Psychology Review, 35'', 6. https://doi.org/10.1007/s10648-023-09724-6 James, W. (1892). The self. In W. James (Ed.), ''Psychology: Briefer course'' (pp. 176–216). Macmillan and Co. https://doi.org/10.1037/11630-012 Lietz, R., Whittaker, S., & Su, N. M. (2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. D., & Horwood, M. S. (2019). The centrality of academic self-concept to motivation and learning. In K. A. Renninger, & S. E. Hidi (Eds.), ''The Cambridge handbook of motivation and learning'' (pp. 36–62). Cambridge University Press. https://doi.org/10.1017/9781316823279.004 Mason, T. B., Smith, K. E., Engwall, A., Lass, A., Mead, M., Sorby, M., Bjorlie, K., Strauman, T. J., & Wonderlich, S. (2019). Self-discrepancy theory as a transdiagnostic framework: A meta-analysis of self-discrepancy and psychopathology. ''Psychological Bulletin, 145''(4), 372–389. https://doi.org/10.1037/bul0000186 Morris, L. S., Grehl, M. M., Rutter, S. B., Mehta, M., & Westwater, M. L. (2022). On what motivates us: A detailed review of intrinsic v. extrinsic motivation. ''Psychological Medicine, 52''(10), 1801–1816. https://doi.org/10.1017/S0033291722001611 Nurra, C., & Oyserman, D. (2018). From future self to current action: An identity-based motivation perspective. ''Self and Identity, 17''(3), 343-364. https://doi.org/10.1080/15298868.2017.1375003 Oyserman, D. (2015). Identity-based motivation. ''Emerging Trends in the Social and Behavioral Sciences, 38'', 1–11. https://doi.org/10.1002/9781118900772.etrds0171 Oyserman, D. (2024), Identity-based motivation and the motivational consequences of difficulty. ''Social and Personality Psychology Compass, 18'', e70028. https://doi.org/10.1111/spc3.70028 Ryan, R. M., & Deci, E. L. (2020). Intrinsic and extrinsic motivation from a self-determination theory perspective: Definitions, theory, practices, and future directions. ''Contemporary Educational Psychology, 61'', 101860. https://doi.org/10.1016/j.cedpsych.2020.101860 Ryan, R. M., & Deci, E. L. (2022). Self-determination theory. In F. Maggino (Ed.), ''Encyclopedia of quality of life and well-being research'' (pp. 1–7). Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] fsuz62cy0zhs6dej4ha9l952tmp4zh6 2835640 2835639 2026-10-03T06:56:45Z U3253363 3106362 /* How self-concept can undermine motivation */ Managed to remove the added title of "quiz" 2835640 wikitext text/x-wiki {{title|Self-concept and motivation:<br>How does self-concept relate to motivation?}} == Overview == {{RoundBoxTop|theme=3}} [[File:Sinclair Swimming.JPG|right|thumb|180px|'''Figure 1'''. Ash is a competitive swimmer and swims daily.]] ;Scenario Ash is a swimmer training for a national competition. Being "a swimmer" is the first thing Ash thinks of when she’s asked to describe herself and she’s eager to succeed in the upcoming competition. Ash wakes up at 4:00 a.m. every morning to swim four kilometres (see Figure 1). Her friend Riley is always surprised by how Ash wakes up so early and swims so much. Why is Ash able to do this when many would find this a struggle? {{RoundBoxBottom}} [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] is an individual's understanding of who they are and how they feel about themselves (Wehrle & Fasbender, 2018). This includes a complex system of attitudes, beliefs, and judgements about oneself. Self-concept guides people in answering the question “who am I?”. But its importance extends beyond that. Self-concept influences thinking, feelings, and behaviour. Consider self-concept an internal compass which motivates and guides how people navigate the world. This chapter explores how the understanding of oneself influences what people are motivated to do and examines how motivated behaviour can shape self-concepts. Understanding this reciprocal relationship can help people develop motivation around meaningful identities while reducing the risk that failure becomes a threat to personal worth. {{RoundBoxTop|theme=12}} '''Focus questions''' *What is self-concept? *How does self-concept influence motivation? *How does motivation shape self-concept?{{RoundBoxBottom}} == What is self-concept? == Self-concept is a cognitive framework containing a system of self-knowledge (Wehrle & Fasbender, 2018). It is related, but unique to many of the popular “self-” constructs like [[self-esteem]], [[w:Self-efficacy|self-efficacy]], and [[w:Self-image|self-image]]. Although each self-concept-relevant theory approaches the construct differently, several common properties emerge and provide the foundation for understanding its motivational effects (see Table 1). [[wikipedia:William_James|James]] (1892) conceptualised the self as having two separate parts: “I”, the subjective thinker and “knower”, and “me”, the material, social, and “known” self. Cooley's (1902) [[wikipedia:Looking-glass_self|looking-glass self theory]] later emphasised people as social beings, theorising that self-concept is a reflection of other peoples' responses and evaluations. Identity-based motivation theory (Oyserman, 2024) and the multidimensional model (Marsh et al., 1992) further develop that self-concept is multifaceted and dynamic, with different parts of the self influencing different behaviours. Together, these theories explain several mechanisms linking self-concept and motivation (see Table 1). '''Table 1''' ''Self-Concept Properties and Their Significance for Motivation'' {| class="wikitable" style="margin: auto;" !Property !Theory !Significance for motivation |- |'''Descriptive  ''' |James |Identity beliefs influence goals and expectations |- |'''Socially developed''' |The looking glass self, James |Social feedback influences which behaviours are pursued |- |'''Dynamic''' |Multidimensional model, identity-based motivation theory |Self-concept changes situationally and with experience |- |'''Multidimensional''' |Multidimensional model |Different aspects of the self motivate different behaviours |- |'''Hierarchical  ''' |Multidimensional model |General and specific domains of the self influence each other and different behaviours |} == How does self-concept influence motivation? == Like self-concept, motivation is multidimensional and central to human behaviour (Bandhu et al., 2024). This creates an interesting intersection of properties; self-concept and motivation both influence which goals people pursue and responses to difficulty. This section explores how self-concept influences motivation.   === How the self directs and energises behaviour === Higgins' (1987) [[w:Self-discrepancy_theory|self-discrepancy theory]] proposes that people have three representations of the self: the actual (who one is), ideal (who one wants to become), and ought self (who one believes others want them to be; see Table 2). Ideal and ought selves set standards for one to strive toward, laying a framework for goal direction. Discrepancies between parts of the self cause emotional vulnerabilities which also generate motivation (Table 2). Actual/ideal self-discrepancies result in dejection-related emotions (e.g. frustration, dissatisfaction; Higgins, 1987). Actual/ought self-discrepancies result in agitation-related emotions (e.g. anxiety, guilt). The greater the discrepancy, the greater the emotional discomfort and thus motivation to reduce the discrepancy (Higgins, 1989). Meta-analytic findings support that greater discrepancies result in more negative emotions (Mason et al., 2019). However, Mason et al. (2019) found that actual/ideal and actual/ought self-discrepancies both correlated with dejection- and agitation-related emotions, contrary to self-discrepancy theory's proposition. Further, while the negative emotions resulting from discrepancies are proposed to motivate people to change behaviour and reduce discrepancies, MacIntyre and Vincze (2017) suggest positive emotions have greater motivating strength than negative emotions. This suggests that while self-concepts provide frameworks for goal selection and energise behaviour, self-discrepancies may not be the most significant emotional sources of motivation. '''Table 2''' ''Dimensions of the Self and Emotional Effects of Discrepancies'' {| class="wikitable" !Self dimension !Examples !Emotional effect of discrepancy with actual self |- |Actual |Daily behaviours, beliefs | - |- |Ideal |Dreams, aspirations |Dejection-related emotions |- |Ought |Duties, obligations |Agitation-related emotions |}[[w:Regulatory_focus_theory|Regulatory focus theory]] (Higgins, 1997) expands self-discrepancy theory to suggest different self-representations produce different motivational orientations. Ideal-self goals create a promotion focus, whereby one is motivated to seek positive outcomes (see Table 3). Ought-self goals create a prevention focus, whereby one is motivated to avoid negative outcomes. These regulatory focuses both positively correlated with intrinsic motivation in Wang et al.'s (2021) study of adolescent task engagement. However, promotion focuses positively correlated with creativity and an innovative cognitive style, whereas prevention focuses negatively correlated with an innovative cognitive style and was characterised by error avoidance. This shows that while both promotion and prevention orientations generate self-sustained motivation, they result in different approaches to tasks. '''Table 3''' ''Self-concept's Motivational Mechanisms'' {| class="wikitable" |+ !Self-discrepancy !Motivational focus !Motivational mechanism !Task approach |- |Actual-ideal |Promotion |Seek positive outcomes |Creativity, innovation |- |Actual-ought |Prevention |Avoid negative outcomes |Reduced innovation, error avoidance |} Oyserman’s (2024) [[wikipedia:Identity_based_motivation|identity-based motivation theory]] provides an alternative explanation to how self-concept directs behaviour. The theory proposes that situations influence which aspects of self-concept are active. Active identities change interpretations of difficulty and thus influence motivation. When behaviour is congruent with one’s active identity, challenges in completing that behaviour are interpreted as evidence the behaviour is important (Oyserman, 2015). This is because identity-congruent behaviours provide information about who one is and who one wants to become, making difficulty meaningful rather than discouraging (Oyserman, 2024). Incongruent actions that have the same challenges are interpreted as pointless. Together, this means that identities mediate motivation and help goal selection through the process of difficulty interpretation (Oyserman, 2024). {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash is finding her long training sessions for the upcoming swimming competition difficult. However, when she's training, her future-self-identity as an Olympic swimmer is active and helps her interpret this as a meaningful challenge. For her non-swimmer friend, Riley, these difficulties may be interpreted as evidence swimming is not for them.{{RoundBoxBottom}} Illustrating the motivating role of the self-concept within this framework, Nurra and Oyserman (2018) found that motivation to pursue future identity-relevant goals in school children, and subsequent achievement, was influenced by how children thought about their future selves. Students were exposed to prompts that helped them think of their adult self as either near or far. Students in the adult-self-is-near group achieved significantly higher final grades than the adult-self-is-far group. These findings indicate that closely identifying with one’s future self and viewing that future self as attainable can motivate goal attainment. Together, these results support identity-based motivation theory and detail that self-concept drives behaviour by defining desirable outcomes and shaping how challenges and opportunities are interpreted. Together, self-discrepancy, regulatory focus, and identity-based motivation theories suggest that a person's perceived distance from a valued self can create motivational pressure to change and active identities influence interpretations of challenges. These states modify goal engagement and motivation. In all three theories, self-concept is central to motivation, driving goal selection as well as goal engagement. Thus, what motivates each person, and how they subsequently behave, are shaped by their self-concept. === How self-concept can undermine motivation === The theories and research discussed so far optimistically show how self-concept facilitates motivation. However, self-concept can also undermine motivation through self-worth contingencies. [[wikipedia:Contingent_self-esteem#Self-esteem_and_contingent_self-worth|Self-worth contingencies]] refer to the standards one feels they must meet to have value as a person (Showers et al., 2015). For individuals who have highly contingent self-worth, goal attainment is closely tied to their self-concept, meaning failure threatens their self-concept. Contingent self-worth motivates pursuit of success for domains that are important for one's self-worth but can result in avoidant behaviour and strong responses to failure. {{RoundBoxTop|theme=11}} <quiz display=simple> '''What are self-worth contingencies?''' |type="()"} - How people feel about themselves + Standards people feel they must meet to have value - How people compare themselves to others </quiz>{{RoundBoxBottom}} Lietz et al. (2026) qualitatively studied 17 participants' identity-shaped goals and found that failing to achieve these goals resulted in negative and severe emotional responses. Further, participants tended to attribute failure to personal shortcomings, because of how closely held the goal was to their self-concept. This resulted in reduced motivation to reattempt, or blind pursuit of the same goal without re-examining the approach. This is consistent with [[Motivation and emotion/Book/2024/Attribution theory and emotion|attribution theory]] (Weiner, 1985), which suggests that causal thinking processes, whereby one attributes outcomes to stable, uncontrollable factors, result in low motivation. Attributional styles affect self-evaluations and thus can undermine motivation when failure is attributed to stable aspects of the self (Bandhu et al., 2024). This can cause maladaptive behaviours to emerge. One such response to the threat of failure is self-handicapping. [[w:Self-handicapping|Self-handicapping]] involves creating obstacles to one's own goal so that failure cannot be self-attributed (Török & Szabó, 2018). A handicap acts as a plausible external explanation for an outcome other than the self. This means that a handicapper is willing to increase the probability of failure to protect their self-concept (Coudevylle et al., 2020). Therefore, self-concept can impact motivation by hindering goal pursuit. {{RoundBoxTop|theme=3}}'''Case study comprehension check''' Ash cares a lot about succeeding in her swimming competition. To protect her self-concept as a successful swimmer, Ash may feign a mild illness at the competition so that any failure is attributed to her illness, rather than her competence. {{RoundBoxBottom}} [[File:Multidimensional_model_of_self-concept_diagram.png|thumb|536x536px|'''Figure 2.''' The hierarchical multidimensional model of self-concept (Marsh et al., 2019).]] Illustrating the causes of self-handicapping, Schwinger et al. (2022) found a medium negative correlation between general self-evaluations and self-handicapping and a medium positive correlation between fear of failure and self-handicapping. With more negative general self-evaluations and greater fear of failure, people are more likely to self-handicap, highlighting self-concept as an precursor to self-handicapping. However, in this study, general, rather than domain-specific measures were used. According to the multidimensional model, self-concept is a hierarchical, multi-faceted construct, and more specific self-concepts are more closely related to actual behaviour (see Figure 2; Marsh et al., 2019). Accordingly, future research should investigate domain-specific measures of self-concept and fear of failure to illustrate their direct influence on self-handicapping and task motivation. Self-handicapping demonstrates that self-concept-driven motivation is not always directed towards achieving valued outcomes. Protecting self-worth can become more important than goal engagement. Together, this research suggests self-concept can act both as a motivational resource, and a motivational vulnerability. {{RoundBoxTop|theme=9}}'''Intervention application: Reduce self-threat''' Self-handicapping shows that holding success as a contingency for self-worth can be damaging (Showers et al., 2015). This means interventions should reduce how much self-worth depends on outcomes, by promoting self-compassion, diverse positive self-concepts, and growth-oriented interpretations of failure to improve motivation and wellbeing.{{RoundBoxBottom}} == How does motivation shape self-concept? == The research and theories discussed so far have detailed how self-concept directs and motivates behaviour. The following section explores how, and whether, this relationship works in reverse.   === How internalising motivation shapes self-concept === [[File:Self-determination theory psychological needs.jpg|thumb|450x450px|'''Figure 3.''' Self-determination theory proposes that autonomy, competence, and relatedness support autonomous motivation (Ryan & Deci, 2020).]][[Self-determination theory]] (Ryan & Deci, 2020) proposes motivation exists on a continuum of self-determination, ranging from externally to increasingly internally regulated behaviour. The continuum consists of external, introjected, identified, integrated, and intrinsic regulation (Table 4). These motivation types are influenced by three innate needs: competence, autonomy, and relatedness (see Figure 3; Ryan & Deci, 2020). Understanding the progression of these needs and motivation types can provide insight into how motivational processes influence self-concept. Intrinsic motivation refers to autonomous behaviours completed for enjoyment (see Table 4; Morris et al., 2022). Extrinsic motivation involves engaging in behaviour for an outcome separable from the activity itself (Morris et al., 2022). Its regulation ranges from externally controlled to personally valued and fully integrated. How self-determined people's motivations are depends on the extent to which one has internalised the values, beliefs and perceptions of the behaviour into their self-concept. Meeting the needs of competence, autonomy and relatedness supports motivations to become progressively internalised, and behaviour increasingly self-endorsed (Chiu, 2023). However, it should be noted that this process does not necessarily occur in a fixed sequence. This illustrates how self-concept and motivation reciprocally form one another. The degree to which one identifies with a behaviour shapes their motivation to engage with the behaviour; when needs are met, motivated behaviours and psychological processes shape the self-concept. This internalisation process helps explain how Ash came to identify as a swimmer (see Table 4).   '''Table 4''' ''Applying Self-Determination Theory to Ash's Motivation and Self-Concept Development'' {| class="wikitable" style="margin: auto;" !Type of motivation !Ash's circumstances   !Mechanism |- |External regulation |Ash received a lollipop after swimming lessons.   |Non-self-determined drive.   |- |Introjected regulation |Ash wants her parents to think she is dedicated to swimming. |Partially internalised extrinsic motivation, low self-determination.   |- |Identified regulation |Swimming becomes important to Ash as she starts feeling like a good swimmer (competence) and drives herself to training (autonomy).   |More internalised and self-determined. |- |Integrated regulation |Ash starts to identify as a swimmer and with her swim team (relatedness).   |Fully integrated into self-concept. Highly self-determined but performed for valued outcome not enjoyment. |- |Intrinsic motivation |After the swimming competition, Ash continues swimming for the joy of it.   |Fully integrated into self-concept. Fully autonomous motivation for enjoyment. |} Chiu (2023) provided compelling evidence for how motivation shapes self-concept within the context of self-determination theory. In their study, 342 Hong Kong high school students completed STEM activities taught using a standard teaching approach or a self-determination theory teaching method, focusing on supporting needs of autonomy, competence and relatedness (see Figure 3). Self-report measures reflected that the needs-supporting teaching approach increased students' feelings of autonomy, competence and relatedness; and, consequently, increased STEM identity and interest. This supports that fulfilling core psychological needs promotes more autonomous motivation and develops one's identity. Further supporting this relationship, Yip et al.'s (2024) longitudinal study of 236 anti-poverty activists who completed self-reported social attitudes and personality measures at two timepoints found that increases in self-determined motivations positively correlated with group identification. However, non-self-determined motivations, such as external regulation, did not correlate with group identification. This suggests that self-determined motivation facilitates the incorporation of behaviour and group membership into one's self-concept.   Together, self-determination theory suggests self-concepts develop as values and behaviours are progressively internalised. Ash was not initially motivated because she identified as a swimmer. Her repeated engagement in swimming and fulfilment of psychological needs gradually transformed swimming into part of her identity. This demonstrates that motivation can be a mechanism through which self-concept develops and is not simply a motivating force. {{RoundBoxTop|theme=11|title=Quiz}} <quiz display=simple> '''Which statement best reflects how motivation shapes self-concept?''' |type="()"} - People are only motivated to engage in behaviours that are part of their self-concept. - Self-concept does not change through motivational processes. + Repeated engagement with a behaviour and satisfaction of psychological needs gradually makes a behaviour part of one's self-concept. </quiz>{{RoundBoxBottom}} === The cyclical process of motivation and self-concept development === The reciprocal effects model proposes that self-concept and achievement reinforce one another over time (Marsh et al., 2018). People with stronger self-concepts tend to achieve more highly, while achievement experiences subsequently shape self-concept. Motivation plays an important role in this process, influencing how people behave toward achievement experiences. There is a growing body of research investigating this model. For example, Sewasew et al.'s (2018) longitudinal study with 2,342 German high schoolers demonstrated mathematics self-concept was positively correlated with mathematics achievement. Their study highlighted a mutually dependent relationship between academic self-concept and achievement, each mediated by motivation. However, Garn and Shen's (2015) longitudinal study of 329 participants' self-concept and motivation for exercise found that the psychological needs posited by self-determination theory (Ryan & Deci, 2022) did not predict physical self-concept at follow-ups. Physical self-concept was only shown to shape, rather than be shaped by exercise motivation. This result conflicts with the reciprocal effects model and Sewasew et al.'s (2018) results, which may be a reflection of the different self-concept domains examined, mathematics versus physical self-concept. This may indicate that reciprocal effects are domain specific; the multidimensional model suggests academic and physical self-concepts function independently, meaning findings from one domain may not generalise to another (Marsh et al., 2019). Thus, future research should directly compare the reciprocal relationship between self-concept and motivation within different domains. Similarly, Sorjonen et al. (2024) reanalysed meta-analytic results which had provided evidence for the reciprocal effects model. Sorjonen et al. (2024) found that conclusions about the longitudinal relationship between achievement and academic self-concept changed depending on the statistical model used. Hübner et al.'s (2023) support for the reciprocal effects model was also inconsistent between statistical models. This may indicate a spurious correlation that undermines the credibility of the reciprocal effects model. Taken together, these findings suggest that achievement can shape self-concept, but the strength and direction of this relationship may depend on the domain being examined and the methodological approach used. Although the reciprocal effects model remains influential, current evidence indicates that the relationship between motivation, achievement, and self-concept is more complex than originally proposed. == Conclusion == Self-concept is a multidimensional and socially developed framework of beliefs about oneself which acts as a guiding compass. Self-concept shapes motivation by determining which goals feel meaningful, attainable, and identity-consistent through processes of self-discrepancy, difficulty interpretation and self-worth contingencies. Self-discrepancy theory explains how distance from valued selves can create motivational pressure for change. Identity-based motivation theory explains how active identities change the meaning of difficulty. Whereas self-worth contingencies can undermine motivation. Together, these theories detail that self-concept influences goal direction and engagement through social, emotional and cognitive processes. These theories provide insight that motivation-improving interventions should consider targeting how people understand themselves rather than solely targeting behaviours. Motivation may also influence self-concept. Self-determination theory details that motivation and subsequent behaviours form self-concept through internalisation processes when psychological needs are met. However, inconsistent support for the reciprocal effects model indicates that the extent to which motivation reshapes self-concept may vary across domains and statistical models. {{RoundBoxTop|theme=3}} '''Take-home message''' With self-concept and motivation inherently linked, who people believe they are shapes the goals they pursue, while motivations, behaviours, and experiences continually reshape who people become. {{RoundBoxBottom}} {{RoundBoxTop|theme=11|title=Quiz}}<quiz display=simple> {'''Chapter comprehension check quiz:''' Ash wakes up at 4:00 a.m. each day to train because being a swimmer is a central part of her identity. According to the chapter, what best explains Ash's motivation? |type="()"} +Self-concept acts as an internal compass that guides goals and behaviour. -Motivation is caused only by external rewards and punishment. -Self-concept remains fixed and has little influence on actions. </quiz> {{RoundBoxBottom}} == See also == * [[wikipedia:Self-concept|Self-concept]] (Wikipedia) * [[Motivation and emotion/Book/2019/Self-concept clarity|Self-concept clarity]] (Book chapter, 2019) * [[Motivation and emotion/Book/2014/Self-efficacy and motivation|Self-efficacy and motivation]] (Book chapter, 2014) == References == {{Hanging indent|Bandhu, D., Mohan, M. M., Nittala, N. A. P., Jadhav, P., Bhadauria, A., & Saxena, K. K. (2024). Theories of motivation: A comprehensive analysis of human behavior drivers. ''Acta Psychologica, 244'', 104177. https://doi.org/10.1016/j.actpsy.2024.104177 Chiu, T.K. (2023). Using self-determination theory (SDT) to explain student STEM interest and identity development. 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(2026). ‘It meant something about who I am’: What the lived experience of failure teaches us about designing for goal setting. ''Behaviour & Information Technology,'' 1–27. https://doi.org/10.1080/0144929X.2026.2711023 MacIntyre, P. D., & Vincze, L. (2017). Positive and negative emotions in motivation for second language learning. ''Studies in Second Language Learning and Teaching, 7''(1), 61–88. https://doi.org/10.14746/ssllt.2017.7.1.4 Marsh, H. W., Byrne, B. M., & Shavelson, R. J. (1992). A multidimensional, hierarchical self-concept. In T. M. Brinthaupt & R. P. Lipka (Eds.), ''The self: Definitional and methodological issues'' (pp. 44–95). State University of New York Press. Marsh, H. W., Martin, A. J., Yeung, A. S., & Craven, R. G. (2018). Chapter 6: Competence self-perceptions. In A. J. Elliot, C. S. Dweck, & D. S. Yeager (Eds.), ''Handbook of competence and motivation: Theory and application'' (pp. 85–115). Guilford Publications. Marsh, H. W., Seaton, M., Dicke, T., Parker, P. 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Springer, Cham. https://doi.org/10.1007/978-3-319-69909-7_2630-2 Schwinger, M., Trautner, M., Pütz, N., Fabianek, S., Lemmer, G., Lauermann, F., & Wirthwein, L. (2022). Why do students use strategies that hurt their chances of academic success? A meta-analysis of antecedents of academic self-handicapping. ''Journal of Educational Psychology, 114''(3), 576–596. https://doi.org/10.1037/edu0000706 Seaton, M., Parker, P., Marsh, H. W., Craven, R. G., & Yeung, A. S. (2014). The reciprocal relations between self-concept, motivation and achievement: Juxtaposing academic self-concept and achievement goal orientations for mathematics success. ''Educational Psychology, 34''(1), 49–72. https://doi.org/10.1080/01443410.2013.825232 Sewasew, D., Schroeders, U., Schiefer, I. M., Weirich, S., & Artelt, C. (2018). Development of sex differences in math achievement, self-concept, and interest from grade 5 to 7. ''Contemporary Educational Psychology, 54'', 55–65. https://doi.org/10.1016/j.cedpsych.2018.05.003 Showers, C. J., Ditzfeld, C. P., & Zeigler-Hill, V. (2015). Self-concept structure and the quality of self-knowledge. ''Journal of Personality, 83''(5), 535-551. https://doi.org/10.1111/jopy.12130 Sorjonen, K., Ingre, M., Melin, B., & Nilsonne, G. (2024). Questioning the reciprocal effects model of academic self-concept and achievement: A reanalysis of a meta-analysis of longitudinal studies and a simulation. ''Sage Open, 14''(4). https://doi.org/10.1177/21582440241292826 Török, L., & Szabó, Z. P. (2018). The theory of self-handicapping: Forms, influencing factors and measurement. ''Ceskoslovenska Psychologie, 62''(2), 173-188. Wang, L., Cui, Y., Wang, X., Wang, J., Du, K., & Luo, Z. (2021). Regulatory focus, motivation, and their relationship with creativity among adolescents. ''Frontiers in Psychology, 12'', 666071. https://doi.org/10.3389/fpsyg.2021.666071 Wehrle, K., & Fasbender, U. (2018). Self-concept. In Zeigler-Hill, V., & Shackelford, T. (Eds.), ''Encyclopedia of personality and individual differences'' (pp. 1–5). Springer, Cham. https://doi.org/10.1007/978-3-319-28099-8_2001-1 Weiner, B. (1985). An attributional theory of achievement motivation and emotion. ''Psychological Review, 92''(4), 548–573. https://doi.org/10.1037/0033-295X.92.4.548 Yip, L., Thomas, E. F., Amiot, C., Louis, W. R., & McGarty, C. (2024). Autonomous motives foster sustained commitment to action: Integrating self-determination theory and the social identity approach. ''Personality and Social Psychology Bulletin, 50''(5), 750–765. https://doi.org/10.1177/01461672221148396 }} == External links == * [https://rickhanson.com/being-well-podcast-self-concept-the-secret-to-changing-who-you-are/ Being well podcast: Self-concept: The secret to changing WHO you are] (Being Well Podcast) * [https://www.youtube.com/watch?v=Nck7lP-6C2g Unlocking success: The power of self-efficacy and self-concept] (TEDx Talks) * [https://www.ted.com/talks/anil_ananthaswamy_where_does_your_sense_of_self_come_from_a_scientific_look Where does your sense of self come from? A scientific look] (TEDx Talks) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-concept]] [[Category:Motivation and emotion/Book/2026]] [[Category:Motivation and emotion/Book/Motivation]] bwjiempq711hvxoangimogim47x5o30 0 331068 2835517 2835485 2026-10-02T14:21:57Z Flickstar888 3106813 Added an image for self-efficacy 2835517 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for over a year, and both applied for a promotion within the team recently, Jon was successful and Sam missed out. Other team members have noticed lately that Sam is putting Jon down in meetings and excluding him from their team lunches. The situation appears to be escalating by the day. This is an example of malicious envy. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> hvllwogwkt597inw8fpnm3fz54qy6em 2835589 2835517 2026-10-02T23:31:49Z Flickstar888 3106813 /* References */ 2835589 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for over a year, and both applied for a promotion within the team recently, Jon was successful and Sam missed out. Other team members have noticed lately that Sam is putting Jon down in meetings and excluding him from their team lunches. The situation appears to be escalating by the day. This is an example of malicious envy. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> drbczgqd7pogza8574eebc2fojvazmq 2835590 2835589 2026-10-02T23:51:39Z Flickstar888 3106813 Added a case study 2835590 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for over a year, and both applied for a promotion within the team recently, Jon was successful and Sam missed out. Other team members have noticed lately that Sam is putting Jon down in meetings and excluding him from their team lunches. The situation appears to be escalating by the day. This is an example of malicious envy. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 7tlwusm7c0pnbd3ib8dyxqqtmurrjm3 2835591 2835590 2026-10-03T00:05:30Z Flickstar888 3106813 Added an image to the case study 2835591 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for over a year, and both applied for a promotion within the team recently, Jon was successful and Sam missed out. Other team members have noticed lately that Sam is putting Jon down in meetings and excluding him from their team lunches. The situation appears to be escalating by the day. This is an example of malicious envy. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> dfp7t9fjjrw7cf8ss9fyg1eqkfqpwl6 2835592 2835591 2026-10-03T00:17:47Z Flickstar888 3106813 Removed image as I prefer text only 2835592 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for over a year, and both applied for a promotion within the team recently, Jon was successful and Sam missed out. Other team members have noticed lately that Sam is putting Jon down in meetings and excluding him from their team lunches. The situation appears to be escalating by the day. This is an example of malicious envy. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> cy8av2rauw9j26dhy702wtsfzd4ja95 2835593 2835592 2026-10-03T00:30:12Z Flickstar888 3106813 worked on the scenario 2835593 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Malicious envy vs benign envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== '''Benign and Malicious Envy''' ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> fesaaoo4gvr4ztvw0qh1j2scqxyj39j 2835594 2835593 2026-10-03T00:34:26Z Flickstar888 3106813 Corrected formatting 2835594 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}Scenario: Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> px5ey8hkki82dslyjmaq837n2pu308l 2835595 2835594 2026-10-03T00:37:46Z Flickstar888 3106813 Changed a word to Bold 2835595 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> msf8umh4f72cy1oc1yeminv3st0ne3l 2835602 2835595 2026-10-03T02:08:53Z Flickstar888 3106813 added a sentence under the header 2835602 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== The following table outlines the difference between benign and malicious envy in the workplace. Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> l0s2xhmum5jjwwar3c2xq1xeewe4htj 2835610 2835602 2026-10-03T04:50:39Z Flickstar888 3106813 worked on research 2835610 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). While psychologists tend to focus on the negative aspects associated with envy, there are many positive aspects. Envy can be described as the resentment you feel when others receive what you deeply desire. We feel envy for a colleague who obtains a promotion or who buys the car we want. This kind of emotion drives our different behaviours. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. This book chapter describes workplace envy, explores the different types of envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== The following table outlines the difference between benign and malicious envy in the workplace. Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> k0tjzx288j5u2sg48x4zeyycnn5cji9 2835620 2835610 2026-10-03T04:58:38Z Flickstar888 3106813 2835620 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). While psychologists tend to focus on the negative aspects associated with envy, there are many positive aspects. Envy can be described as the resentment you feel when others receive what you deeply desire. We feel envy for a colleague who obtains a promotion or who buys the car we want. This kind of emotion drives our different behaviours. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== The following table outlines the difference between benign and malicious envy in the workplace. Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> c9ff4o5vus4jd2ms14mqdijalmnzuvg 2835622 2835620 2026-10-03T05:11:52Z Flickstar888 3106813 /* Role of envy in the workplace */ 2835622 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). While psychologists tend to focus on the negative aspects associated with envy, there are many positive aspects. Envy can be described as the resentment you feel when others receive what you deeply desire. We feel envy for a colleague who obtains a promotion or who buys the car we want. This kind of emotion drives our different behaviours. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Envy plays an important role in workplace behaviours. There are two types of workplace envy: * Benign envy * Malicious envy Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== The following table outlines the difference between benign and malicious envy in the workplace. Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> oppbckjcnxd2lpfnlb8bhh0ykc2dd9s 2835631 2835622 2026-10-03T06:07:04Z Flickstar888 3106813 /* Benign versus malicious envy */ 2835631 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). While psychologists tend to focus on the negative aspects associated with envy, there are many positive aspects. Envy can be described as the resentment you feel when others receive what you deeply desire. We feel envy for a colleague who obtains a promotion or who buys the car we want. This kind of emotion drives our different behaviours. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Benign and malicious envy both start with the same feeling of inferiority although they both lead to different reactions (Crusius et al, 2021). Malicious and benign envy both exist in workplaces. * Malicious and benign envy exist in all workplaces. * Malicious envy - negative behaviours will be described and how they influence the workplace. * Benign envy - positive behaviours will be described and how they influence the workplace. {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}} ==== Benign and Malicious Envy ==== The following table outlines the difference between benign and malicious envy in the workplace. Table 1. {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 3yji0yeqk498qkutnv7wvkedtemqofd 2835632 2835631 2026-10-03T06:16:33Z Flickstar888 3106813 /* Benign versus malicious envy */ 2835632 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). While psychologists tend to focus on the negative aspects associated with envy, there are many positive aspects. Envy can be described as the resentment you feel when others receive what you deeply desire. We feel envy for a colleague who obtains a promotion or who buys the car we want. This kind of emotion drives our different behaviours. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. Envy is the driver for many different workplace behaviours: * competitiveness * unfairness * catalyst for change * another example While the negative effects of envy attract the most attention, there are also positive aspects that can bring a wealth of growth and encouragement to the workplace. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Benign and malicious envy both start with the same feeling of inferiority although they both lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 2zcrlpaw46lc74ryih98nraz980ozvz 2835641 2835632 2026-10-03T07:34:24Z Flickstar888 3106813 worked on overview 2835641 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. The workplace is filled with opportunities for competition and self-comparison. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Benign and malicious envy both start with the same feeling of inferiority although they both lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> jgmqubza68ll0mm2wi72tn62qzhcr1e 2835642 2835641 2026-10-03T07:41:14Z Flickstar888 3106813 /* Benign versus malicious envy */ 2835642 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. The workplace is filled with opportunities for competition and self-comparison. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they both lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> lfo8vaty119a3f1290ij5mtiagwb2oo 2835644 2835642 2026-10-03T07:58:03Z Flickstar888 3106813 /* Overview */ 2835644 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they both lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 7e1lxhxa458jcvcf4epgkj2cc3u6m7q 2835661 2835644 2026-10-03T10:15:43Z Flickstar888 3106813 /* Benign versus malicious envy */ 2835661 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> q10c1l4j46zkl6b0lf8b9x6v5haaypz 2835662 2835661 2026-10-03T10:21:07Z Flickstar888 3106813 /* References */ 2835662 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion will answer the questions. * The take home messages are that envy is an important catalyst for change in the workplace if driven in a positive way. * Healthy competition and influence of envy. * Role of motivation and envy in shaping workplace culture. * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> d1oo0auu9ci6l4f57x7kczzztrzgokb 2835670 2835662 2026-10-03T11:11:01Z Flickstar888 3106813 /* Conclusion */ 2835670 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 3x3xmhkpn8i67a3xop0qjr4uz7u77r7 2835671 2835670 2026-10-03T11:16:38Z Flickstar888 3106813 /* Behaviours caused by self-efficacy? */ 2835671 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above { &nbsp; |type="{}"} Where was Aristotle born? { Stageira } { &nbsp; |type="{}"} Name Aristotle's famous mentor. { Plato } { &nbsp; |type="{}"} { Alexander } was Aristotle's most famous pupil. == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> nihxiw0qwftjjswetbq66wsrzooqujy 2835672 2835671 2026-10-03T11:17:45Z Flickstar888 3106813 removed quiz 2835672 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory was introduced by Festinger in 1954. This model assumes that individuals tend to compare their opinions and/or abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> lqnnkb0vmm7fi7tq3fmxa3xf5sn5ufe 2835673 2835672 2026-10-03T11:19:17Z Flickstar888 3106813 /* Relationship between workplace envy and social comparison theory? */ 2835673 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory assumes that individuals tend to compare their opinions and abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2013/Jealousy|title=Motivation and emotion/Book/2013/Jealousy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-10-02}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 3xtyymf03t9r7u1qtpsjkawqn8vomay 2835675 2835673 2026-10-03T11:43:24Z Flickstar888 3106813 /* External links */ 2835675 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory assumes that individuals tend to compare their opinions and abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=|title=|website=|language=|access-date=}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== The following links provide further background and context to support this book chapter: * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Envy]] [[Category:Motivation and emotion/Book/Organisational psychology]] <references /> 1ozr0eiv2nntq88wsoavow4p094b9ww 2835676 2835675 2026-10-03T11:44:57Z Flickstar888 3106813 2835676 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory assumes that individuals tend to compare their opinions and abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=|title=|website=|language=|access-date=}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== The following links provide further background and context to support this book chapter: * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] s2vboe8kqtijqbfg8jywi2rqbzt4m57 2835677 2835676 2026-10-03T11:48:03Z Flickstar888 3106813 2835677 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any modern workplace employees routinely compare their performance or progress to those around them, these comparisons easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory assumes that individuals tend to compare their opinions and abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=|title=|website=|language=|access-date=}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== The following links provide further background and context to support this book chapter: * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} o0tgattt78c1ap9rrg1lqwj1pmjk9w2 2835678 2835677 2026-10-03T11:49:33Z Flickstar888 3106813 /* Role of envy in the workplace */ 2835678 wikitext text/x-wiki {{title|Envy in the workplace:<br>What role does envy play in workplace behaviour?}} __TOC__ === Overview === {{RoundBoxTop|theme=9}} [[File:Zonaspace-coworking-collaboration.jpg|thumb|320x320px|'''Figure 1'''. Envy can occur in any workplace.]] ;Case study Jillian and Anna work for the same consulting firm and are at the same level. Jillian recently completed a project that received recognition from senior leaders and she continues to be praised about it. Anna and Jillian get along well, although Anna has started feeling resentful towards Jillian (see Figure 1). Anna recently asked Jillian about the project and what tools she used that could possibly assist with her projects. Jillian mentioned she completed a project management course that really changed the way she approaches projects to ensure all milestones are delivered on time and on budget. Anna enrols in the same project management course immediately. This is an example of benign envy. By feeling envy towards Jillian, it has motivated Anna to take action to improve herself. With Anna developing her project management skills, this will impact the team and consulting firm in a positive way encouraging extraordinary results. {{RoundBoxBottom}} [[wikipedia:Envy|Envy]] is a powerful [[emotion]] that drives different behaviours (Yang & Tang, 2021). It can be described as the bitterness or anger you feel when others receive what you deeply or unconsciously desire. In many workplaces, it is human nature for an individual to measure and compare themselves to those that are being recognised or seem more valued than they are. These comparisons can trigger workplace envy, especially when someone feels they are inferior to others. According to Duffy et al (2021), over the last 25 years, there has been growing interest in the phenomenon of workplace envy. While every workplace is filled with opportunities for competition and self-comparison, Liu et al (2021), stated it was important to consider the environment and the timing when looking at workplace envy as it is constantly evolving. Workplace envy occurs across a variety of workplace situations although experiences can vary in intensity and frequency and can transform into behavioural reactions. This book chapter describes workplace envy and discusses the role it plays in workplace behaviour. {{RoundBoxTop|theme=3}} '''Focus questions''' * What is the role of envy in the workplace? * What is the difference between malicious envy and benign envy? * What causes envy and how is it linked to motivation? * What is the relationship between envy and self-efficacy? * What is the relationship between envy and social comparison theory? {{RoundBoxBottom}} == Role of envy in the workplace == There are many causes of envy in the workplace. It can emerge when an employee notices that a colleague has achieved something they desire, such as recognition, a reward, a promotion or an opportunity. It can occur in any workplace where employees routinely compare their performance or progress to those around them, these comparisons can easily activate feelings of envy. Through guidance and intervention, envy can be used as a workplace 'tool' to drive positive behaviour. *Envy can be destructive or can lead to a promotion (Liu et al, 2021) *can drive cultural change *envy can occur when individuals compare themselves to others, they perceive to be more fortunate. == Benign versus malicious envy == Workplace envy can be either benign or malicious. Even though they both start with the same feeling of inferiority, they can lead to different reactions (Crusius et al, 2021). Benign envy creates an urge for people to grow while malicious envy leads to negative thoughts or actions that aims to bring a person down (see Table 1). {{RoundBoxTop|theme=3}}'''Scenario:''' Jon and Sam have worked together for a long time. They both applied for a promotion within the team recently, Jon was successful and Sam missed out. Since then, Sam has started treating Jon unfairly. In meetings, Sam makes negative comments about Jon's ideas. During team lunches, Sam leaves Jon out on purpose. Each day, Sam's behaviour becomes more unfriendly and hurtful. This is an example of malicious envy. Sam is feeling envious of Jon and is trying to harm and put him down. {{RoundBoxBottom}}The following table outlines the difference between benign and malicious envy in the workplace. '''Table 1.''' {| class="wikitable" ! ! Benign Envy ! Malicious Envy |- | Definition | Envy that motivates self‑improvement. | Envy that manifests when someone feels threatened or unable to match another’s achievements. |- | Emotional Response | Encourages growth, motivation and learning. | Creates feelings of resentment, threat, or inferiority. |- | Typical Behaviours | Working harder, learning new skills, seeking growth or development opportunities. | Exclusion, gossip, criticism or subtle sabotage. |- | Impact on Workplace Culture | Strengthens workplace culture through healthy competition and personal development. | Damages trust, teamwork, morale, and can escalate into workplace conflict and create a toxic culture. |- | Outcome | Positive: supports improvement and collaboration. | Negative: undermines relationships and workplace functioning. |} == Relationship between envy and motivation? == Envy is linked to motivation. The relationship between them is: * motivation can drive envy * what motivates positive behaviour * what motivates negative behaviour * how, what and why? == Behaviours caused by self-efficacy? == Self-efficacy is the mental switch that determines whether envy will lead to a positive or negative behaviour. The two types of self-efficacy are: * [[File:Self and environment.png|thumb|'''Figure 3.''' Self-efficacy factors.]]high self-efficacy * low self-efficacy * Example of each of the above == Relationship between workplace envy and social comparison theory? == Social comparison theory assumes that individuals tend to compare their opinions and abilities to others. Feeling threatened in the workplace makes people judge themselves more harshly as they believe they cannot match their colleague's social status or success. {{RoundBoxTop|theme=9}} [[File:Call center (8300442093).jpg|thumb|320x320px|'''Figure 4'''. Social Comparison Theory at play in a Call Centre.]] ;Case study At ReadyCall, everyone's daily call numbers are recorded on a big screen. Ebony is new and she keeps comparing herself to Rick, a senior worker who always has the highest call numbers. Seeing the gap makes Ebony feel stressed and worried she isn't good enough. Ebony's manager explains that Rick learned his skills over many years and wasn't fast when he first started. After hearing this, Ebony feels better, learns some tips from Rick, and becomes more confident. This is an example of social comparison theory, where people judge themselves by comparing their performance to others (see Figure 4). {{RoundBoxBottom}} [[File:Happy-smiley-face-and-sad-2048x1536.jpg|thumb|'''Figure 2.''' Envy can drive positive or negative behaviour in the workplace.]] [[wikipedia:Envy|Envy]] is a powerful [[emotion]]. It can be described as the resentment you feel when others receive what you deeply desire. In many workplaces, it is human nature for people to compare themselves to others. In the workplace, a person will measure and compare themselves against those that are being recognised for their work, those that are promoted and those that seem more valued than they are. These comparisons can trigger envy, especially when someone feels they are inferior to others. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display="simple"> {Benign envy is associated with positive emotions and encourages positive behaviour? |type="()"} + True - False {Malicious envy is associated with negative emotions and encourages negative behaviour? |type="()"} - True + False </quiz> ==Conclusion== Envy plays a significant role in shaping workplace behaviour because employees constantly observe and compare their progress and recognition and opportunities of those around them. These comparisons can trigger envy which is not just a negative emotion but a complex psychological response influenced by self-efficacy, motivation and social comparison processes. These comparisons highlight an individual's insecurities, making envy an important emotional signal rather than a something negative. When employees feel capable of improving themselves, workplace envy becomes a constructive force that encourages learning, effort and healthy competition. However, workplace envy can also shift into a harmful direction when employees feel threatened, insecure or unable to achieve similar success. In these situations, workplace envy may lead to resentment, withdrawal or undermining behaviour which can disrupt teamwork or damage workplace relationships. By understanding these dynamics, workplaces can respond carefully and recognise the important role envy plays in fostering growth. fairness and healthier interpersonal relationships that will effectively create a positive working culture. {{tip|Take-home messages: * Not all workplace envy is negative. * Envy is an important catalyst for change if driven in a positive way. * Envy can motivate staff in a constructive way. }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * Envy<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Motivation_and_emotion/Book/2011/Envy|title=Motivation and emotion/Book/2011/Envy - Wikiversity|website=en.wikiversity.org|language=en|access-date=2026-08-25}}</ref> (Book Chapter, 2011) {{ic|See Tutorial 2 for how to create an internal link to a book chapter) * Jealousy<ref>{{Cite web|url=|title=|website=|language=|access-date=}}</ref> is not the same as envy (Book Chapter, 2013) * [[Self-Efficacy]] (Wikipedia) * [[Motivation and emotion/Book/2010/Aggression in the workplace|Motivation and emotion/Book/2010/Aggression in the workplace - Wikiversity]] * [[Motivation and emotion/Book/2024/Rivalry in the workplace|Motivation and emotion/Book/2024/Rivalry in the workplace - Wikiversity]] {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Crusius J., Thierhoff J., Lange J. (2021). Dispositional greed predicts benign and malicious envy. Personality and Individual Differences, 168, 110361. https://doi.org/10.1016/j.paid.2020.110361 Dong, Xue T., Yang W. Chung, Joon H. Park, and Jeong K. Yun. 2026. “Workplace Envy and Organizational Citizenship Behaviour: The Mediating Effect of Cynicism and the Moderating Effect of Organizational Support.” Canadian Journal of Administrative Sciences / Revue Canadienne des Sciences de l'Administration: e70020. https://doi.org/10.1002/cjas.70020. Duffy, M. K., Lee, K., & Adair, E. A. (2021). Workplace envy. Annual Review of Organizational Psychology and Organizational Behavior, 8, 19–44. https://doi.org/10.1146/annurev-orgpsych-012420-055746 Erdil O., Müceldili B. (2014). The effects of envy on job engagement and turnover intention. Procedia - Social and Behavioral Sciences, 150(1), 447–454. https://doi.org/10.1016/j.sbspro.2014.09.050 Han, S., Zhan, Y., Zhang, L., & Mu, R. (2022). You Have Received More Help than I Did and I Envy You: A Social Comparison Perspective on Receiving Help in the Team. International Journal of Environmental Research and Public Health, 19(14), 8351. https://doi.org/10.3390/ijerph19148351 Jafri M. H. (2020). Moderating role of self-efficacy on benign envy—employee engagement relationship. Business Perspectives and Research, 8(2), 232–243. https://doi.org/10.1177/227853371988745 Köksal, K., Tufan, C., Griffiths, M.D. et al. From perceived workplace envy to deviance: a multi-theoretical model of envy-related workplace behaviors and the moderating role of social courage. Curr Psychol 45, 1052 (2026). https://doi.org/10.1007/s12144-026-09607-3 Lange, J., Paulhus, D. L., & Crusius, J. (2018). Elucidating the Dark Side of Envy: Distinctive Links of Benign and Malicious Envy With Dark Personalities. Personality and Social Psychology Bulletin, 44(4), 601-614. https://doi.org/10.1177/0146167217746340 Li M., Xu X., Kwan H. K. (2021). The antecedents and consequences of workplace envy: A meta-analytic review. Asia Pacific Journal of Management. Advance online publication. https://doi.org/10.1007/s10490-021-09772-y Li M, Qin Y, Le J (2025), "Mentoring matters: decoding the impact of downward envy on interpersonal conflict, workplace ostracism, and displaced aggression". Journal of Managerial Psychology, Vol. 40 No. 1 pp. 67–81, doi: https://doi.org/10.1108/JMP-12-2023-0768 Li, Y., & Wang, S. (2023). "Comparisons are Odious"? - Exploring the Dual Effect of Upward Social Comparison on Workplace Coping Behaviors of Temporary Agency Workers. Psychology research and behavior management, 16, 4251–4265. https://doi.org/10.2147/PRBM.S425946 Liu, H., Geng, J., & Yao, P. (2021). Relationship of Leadership and Envy: How to Resolve Workplace Envy with Leadership-A Bibliometric Review Study. Journal of Intelligence, 9(3), 44. https://doi.org/10.3390/jintelligence9030044 Nakai, A. and Numazaki, M. (2026), The Emotional Nature of Malicious and Benign Envy: Separate Measurements for Emotion and Motivation. Jpn Psychol Res. https://doi.org/10.1111/jpr.12568 Song, Y., & Zhao, Z. (2022). Social Undermining and Interpersonal Rumination among Employees: The Mediating Role of Being the Subject of Envy and the Moderating Role of Social Support. International Journal of Environmental Research and Public Health, 19(14), 8419. https://doi.org/10.3390/ijerph19148419 Su, X., & Chen, C. (2023). The Influence of Workplace Envy on Employees’ Knowledge-Hiding Behavior Based on a Comparative Analysis between Generation Cohorts. Behavioral Sciences, 13(9), 716. https://doi.org/10.3390/bs13090716 Walter CE, Au-Yong-Oliveira M (2022), "An exploratory study on the barriers to innovative behavior: the spiteful effect of envy". Journal of Organizational Change Management, Vol. 35 No. 6 pp. 936–960, https://doi.org/10.1108/JOCM-02-2022-0034 Wilkes, D. (2023) Workplace Envy: Exploring the Effect of Envy on Team Communication and Organizational Information Sharing. Open Journal of Business and Management, 11, 1757-1786. https://doi.org/10.4236/ojbm.2023.114099 Yang C and Tang R (2021) Validating the “Two Faces” of Envy: The Effect of Self-Control. Front. Psychol. 12:731451. https://doi.org/10.3389/fpsyg.2021.731451 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== The following links provide further background and context to support this book chapter: * [https://www.youtube.com/shorts/NwGMsgB9mNg Envy: The Ugliest Emotion? Find out the Truth!] * [https://www.youtube.com/watch?v=1f9IE3CkB4s How to Stop Comparing Yourself to Others] * https://www.ted.com/talks/worklife_with_adam_grant_become_friends_with_your_rivals * [https://ideas.ted.com/do-you-secretly-feel-good-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/ https://ideas.ted.com/do-you-secretly-feel-goo-when-others-stumble-5-ways-to-make-peace-with-this-very-human-emotion/] * [https://www.youtube.com/shorts/3g0lRfiAXzc Why Workplace Envy Is Secretly Power ? - YouTube] {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} jsposn267oqvtsf5jv1d3qcjkzbukzs 0 331092 2835505 2834776 2026-10-02T12:56:08Z U3275909 3106769 2835505 wikitext text/x-wiki {{title|Mindset and stigma:<br>What role do growth versus fixed mindsets play in prejudice and stigma?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Fixed versus growth mindset.png|thumb|Figure 1. Fixed and growth mindsets provide contrasting ways of understanding whether human characteristics can change]] ;Two different reactions Emma has recently started a new job. During lunch, she mentions that she has previously received treatment for a substance-use disorder. One colleague believes that personal characteristics are largely fixed and assumes that Emma will always be unreliable. Another colleague also wonders why Emma did not change her behaviour sooner. The colleagues' reactions demonstrate how beliefs about changeability may influence stigma. A fixed mindset may encourage stable stereotypes, whereas a growth mindset may create hope for change. However, believing that change is possible can also increase blame when a person is perceived as responsible for not changing. This chapter examines these competing pathways (see Figure 1) {{RoundBoxBottom}} This example illustrates a central problem in the psychology of [[wikipedia:Social_stigma|stigma]]; beliefs about change can generate both compassion and condemnation. A fixed mindset treats human attributes a relatively stable, while a growth mindset treats then as capable of development (Molden and Dweck, 2006). Fixed beliefs can make stereotypes appear natural and permanent. Growth beliefs can weaken these assumptions encourage hope and motivate positive contact. However, changeability may be confused with controllability. If observers beliefs that a person could change through effort alone, they may hold that person responsible for not changing (Hoyt & Burnette, 2020){{RoundBoxTop|theme=3}} '''Focus questions''' * What are growth and fixed mindsets, and how are they connected to stigma * How do fixed mindsets contribute to stretyping, essentialism and prejudice * Through what processes growth mindsets reduce prejudice and stigma? * When can growth mindsets reduce prejudice and stigma? * How can mindset research inform effective stigma-reduction interventions {{RoundBoxBottom}} == What are growth and fixed mindsets? == Mindsets, also called implicit theories, are beliefs about whether human attributes are fixed or can develop. The word ''<u>implicit</u>'' does not mean that people are necessarily unaware of these beliefs. Rather, it indicates that mindsets function as informal theories through which people organise social information and explain behaviour (Molden & Dweck, 2006). They influence what people notice, the causes they assign to success and failure, and what they expect to happen next. === Fixed mindsets === A fixed or entity mindset proposes that an attribute reflects an enduring quality. Someone holding a fixed mindset about personality might interpret an unfriendly act as evidence that a person ''is'' unfriendly. A growth or incremental mindset proposes that attributes can develop through learning, experience, strategies, support and changing circumstances. A person with this view may still judge the behaviour negatively, but is more likely to ask what produced it and whether it could change (Molden & Dweck, 2006). Mindsets are not necessarily global personality traits. The same person may believe that intelligence can develop, but that prejudice is permanent, or that recovery from depression is possible while body weight is entirely controllable. This domain specificity is crucial for understanding stigma. A general endorsement of "growth" does not reliably reveal how someone will judge a particular group or condition (Hoyt & Burnette, 2020; Molden & Dweck, 2006). Mindsets influence motivation because expectations about change affect whether action seems worthwhile. If prejudice is seen as fixed, learning about bias or attempting a difficult intergroup conversation may seem pointless. If it is malleable, effort can seem meaningful. Growth beliefs are generally associated with mastery-oriented goals and adaptive responses to setbacks, although effects vary across contexts (Burnette et al., 2013). Mindsets also shape attribution, or the causes assigned to behaviour. Entity theorists tend to favour dispositional explanations, inferring stable traits from actions. Chiu et al. (1997) found that implicit theories of personality were connected with this tendency toward lay dispositions. Incremental theorists are comparatively attentive to processes and situations. This does not mean that growth-minded people ignore personal responsibility. Instead, they are more likely to understand behaviour as developing through an interaction among the person, experience and environment. This distinction connects mindsets with psychological essentialism. Essentialism is the belief that members of a category share a deep, defining essence that produces their observable characteristics. Social categories may consequently appear uniform, natural and sharply separated from one another. A fixed mindset can strengthen essentialism because stable attributes seem to require an enduring internal cause. A growth mindset can weaken it by highlighting variability, development and context. However, fixedness and essentialism are not identical; a characteristic can be stable without being biological, and biological explanations do not always imply absolute permanence. A careful theoretical framework therefore treats mindset, essentialism, controllability and blame as related but distinct constructs (Hoyt & Burnette, 2020, 2025; Levy et al., 1998). '''Table 1 - Comparing fixed and growth mindsets''' {| class="wikitable" !Aspect !Fixed mindset !Growth mindset |- |Core belief |Personal and group characteristics are relatively permanent. |Personal characteristics and attitudes can develop or change. |- |Explanation of behaviour |Emphasises enduring traits and internal characteristics. |Considers learning, circumstances, strategies and social support. |- |Connection with essentialism |May strengthen beliefs that groups possess an inherent and defining essence. |May weaken beliefs that group differences are natural and permanent. |- |Effect on stereotypes |May encourage generalisation and stable stereotypical judgements. |May encourage attention to individual differences and new information. |- |Intergroup expectations |May create pessimism about whether people, prejudice or relationships can improve. |May produce more positive expectations and encourage intergroup contact. |- |Potential benefit |May reduce blame when a characteristic is perceived as uncontrollable. |May reduce prejudice through lower essentialism and greater expectations for change. |- |Potential risk |May promote social avoidance and the belief that stigmatised people cannot change. |May increase blame when people are considered responsible for not changing. |} ==<small>Note: Based on Hoyt and Burnette (2020, 2025), Molden and Dweck (2006), Levy et al. (1998)</small>== Neither label guarantees a compassionate outcome, Fixed beliefs may reduce responsibility, while growth beliefs may become punitive when translated into a demand to change. The crucial issue is the casual and moral meaning attached to change (Hoyt & Burnette, 2020). == How do fixed mindsets sustain prejudice and stigma? == Prejudice refers to negative attitudes toward people because of their group membership, whereas stigma involves social devaluation associated with an attribute, identity or condition. Stigma includes stereotypes, emotional reactions, status loss and discriminatory behaviour. Mindsets contribute to these processes by shaping whether people view differences as defining and whether future improvement seems possible. === Essentialism and stereotyping === Fixed mindsets encourage observers to convert behaviour into identity. When people treat an action as evidence of an unchanging essence, differences within a group receive less attention, and stereotyping becomes easier. Levy et al. (1998) reported across five experiments that entity theories were associated with greater stereotype formation and endorsement. Levy and Dweck (1999) extended this work to children. Children judged unfamiliar schools described through positive or negative behaviours; those encouraged to adopt static conceptions of people formed stronger stereotypical impressions than those given dynamic conceptions. These experiments help establish causality because researchers changed the framework through which participants processed group information. Using unfamiliar groups also reduced the influence of established political attitudes. Nevertheless, artificial school and brief messages are not equivalent to historically marginalised groups. Institutions, media and unequal contact reinforce real-world stereotypes. The findings demonstrate a cognitive mechanism, not proof that a short message can undo entrenched stigma. Essentialism can legitimise inequality by making social arrangements appear inevitable. If poverty is interpreted as evidence of an enduring type of person, structural causes such as labour conditions, housing costs or discrimination become less visible. Similarly, if addiction is regarded as a fixed moral defect, a person's recovery may be discounted, and exclusion may appear sensible. Fixed beliefs can therefore preserve stigma even when overt hostility is socially unacceptable; people may frame avoidance as realism rather than prejudice. '''<big>Pessimistic expectations and avoidance</big>''' Fixed mindsets can also affect intergroup behaviour by producing pessimistic expectations. If people assume that their own bias cannot change, they may anticipate anxiety, failure or appearing prejudiced during an interaction. Avoidance then becomes a protective strategy. Unfortunately, avoiding contact removes opportunities to discover individual variation and experience cooperation, leaving negative expectations untested. Carr et al. (2012) examined this process across eight studies. White participants who viewed prejudice as fixed showed less interest in interracial interaction, diversity activities and efforts to reduce their own prejudice. These patterns occurred beyond participants' measured explicit or implicit prejudice. In laboratory interactions, fixed beliefs were associated with less friendly behaviour, even when participants did not report stronger racial prejudice. The research usefully distinguishes hostility from motivation withdrawal; discriminatory social outcomes can arise because a person expects intergroup contact to go badly, not only because they consciously dislike another group. Neel and Shapiro (2012) similarly found that beliefs about whether racial beliefs is malleable predicated different strategies for difficult interracial encounters. Across four studies, these beliefs were both measured and experimentally manipulated and were related to imagined as well as actual interaction strategies. Fixed beliefs encouraged avoidance-oriented responses, whereas malleable beliefs supported learning-oriented engagement. Using multiple methods strengthens confidence that the findings are not merely an artefact of one questionnaire. However, both program focused largely on White Americans, limiting generalisability across cultures, groups and stigmas. Laboratory interactions also cannot reproduce the risks faced by stigmatised people, whose avoidance after discrimination may represent self-protection rather a motivational deficit. '''A self-reinforcing cycle''' Together, essentialism and avoidance can form a cycle. A fixed belief encourages categorial judgement; that judgement creates negative expectations; expectations motivate avoidance; and avoidance prevents corrective experience. The absence of positive contact then seems to confirm that groups are fundamentally different. Mindsets therefore influence both cognition and motivation. They organise stereotypes while also determining whether people approach situations capable of challenging them (Carr et al., 2012; Levy et al., 1998; Neel & Shapiro, 2012). == How can growth mindsets reduce prejudice and stigma? == Growth mindsets interrupt this cycle by presenting people, attitudes and relationships as capable of development. Their benefits operate through at least two pathways; reducing essentialism and creative constructive expectations about engagement. These processes are related but should not be collapsed. A person might recognise variation within a stigmatised group without wanting contact, or seek contact while retaining stereotypical beliefs (Hoyt & Burnette, 2025). === Reduced essentialism and encouraging individuation === If attributes can develop, one action provides less evidence of an enduring essence. Observers have a reason to consider time, context and individual variation. Growth beliefs can consequently encourage <u>individuation</u> - judging a person using information specific to them rather than treating them as interchangeable with a category. Emma's history of substance use then becomes one part of a changing life rather than a permanent identity (Levy et al., 1998; Molden & Dweck, 2006). Levy and colleagues findings support this pathway but also reveal an important boundary. Dynamic theories did not deny group patterns; they reduced the generalisation of limited information into stable traits (Levy et al., 1998; Levy & Dweck, 1999). The goal is therefore not to ignore social identity or inequality, but to reject the idea that membership determines every person's character or potential. Growth beliefs about poverty have likewise been associated with lower stigma toward people experiencing poverty, illustrating how attention can shift from an allegedly fixed type of person toward changeability circumstances and pathways (Hoyt & Burnette, 2025). Still, reduced stigma may depend on what participants think produces change. If change is imagined as access to education, secure housing and fair employment, growth implies social possibility. If it is imagined only as willpower, the same message may intensify responsibility. This distinction foreshadows the double-edged effect discussed below. === Supporting positive intergroup contact === Growth mindsets can also change emotional and motivational expectations. Difficult intergroup interactions often involve anxiety about rejection, embarrassment or causing offence. Believing that people and prejudice can change makes an imperfect interaction less diagnostic; discomfort becomes a manageable part of learning rather than proof that future contact will fail (Carr et al., 2012; Hoyt et al., 2024). Carr et al. (2012) found that malleable beliefs about prejudice increased interest in interracial interactions and prejudice-reduction activities. Neel and Hapiro (2012) showed that these beliefs predicted more learning-oriented strategies, such as seeking feedback, rather than simply avoiding situations where bias might be revealed. More recently, Hoyt et al. (2024) found that growth mindsets about people were linked with stronger interracial contact intentions among White Americans through more positive expectations and lower anxiety. This proposed mediation is theoretically coherent; beliefs about human development alter anticipated outcomes, which then influence approach motivation. This evidence is promising, but intentions are not sustained relationships. Positive contact depends on cooperation, institutional support and whether stigmatised people feel safe. Encouraging contact without preparing participants to listen may place the intervention's burden on those affected by stigma. Growth-oriented initiatives therefore require appropriate norms and structures (Hoyt et al., 2024). == When can growth mindsets increase stigma? == Popular discussions often portray growth mindsets as uniformly positive. Stigma research shows why this conclusion is unsafe. Changeability describes whether change is possible; controllability describes how much personal choice determines an outcome. The concepts can be psychologically confused. When people hear that weight, poverty, addiction or mental health can change, they may infer that individuals can change whenever they choose. Failure to change then appears to reveal laziness, weak character or insufficient motivation (Hoyt & Burnette, 2020; Hoyt et al., 2017). === The stigma-asymmetry model === Hoyt et al. (2017) described this tension through the obesity stigma asymmetry model. Across three experiments, participants received messages presenting obesity as a disease or weight as changeable. Changeability messages could reduce essentialist thinking while simultaneously increasing blame, and these indirect pathways had opposing implications for anti-fat prejudice. Research on weight beliefs similarly found that mindset messages predicted anti-fat attitudes and body shame through casual attributions (Burnette et al., 2017). The value of this work is that it avoids asking only whether a message has a positive or negative total effect. An apparently weak overall result can conceal two strong mechanisms operating in opposite directions. Reduced essentialism may lower prejudice while increased blame raises it. This explains why a growth message can improve one attitude but fail to improve, or even worsen, another (Hoyt et al., 2017). Experimental health messages simplify complex conditions, and immediate responses responses do not show whether attitudes endure or guide behaviour. Weight stigma also cannot represent race, disability, mental illness or substance use. Nevertheless, the model warns that a claim which creates hope can also create a moral obligation to change (Hoyt & Burnette, 2020; Hoyt et al., 2017). '''The costs of compulsory optimism''' Growth language becomes harmful when it suggests that effort guarantees success. People differ in access to treatment, time, money, safety and supportive relationships. Symptoms may fluctuate, discrimination may restrict opportunities, and some characteristics do not require correction. Telling a person that '''''"anything is possible if you try"''''' can erase these realities and turn unequal outcomes into evidence of personal failure (Hoyt & Burnette, 2020) This risk is especially important when the target of change is ambiguous, An intervention about disability might encourage institutions to improve accessibility, or imply that disabled people should overcome their limitations. A campaign about prejudice might motivate non-stigmatised people to revise biased beliefs, or pressure minority-group members to appear more acceptable. Ethican growth messaging must ask, '''''"Who is being asked to change, and for whose benefit?"''''' Fixed beliefs can occasionally reduce blame by communicating that a condition is not freely chosen. Biological explanations of mental illness, for example, may reduce blame but can also increase perceptions of dangerousness and social distance (Kvaale et al., 2013). This trade off reinforces the broader argument, reducing one component of stigma does not guarantee an overall improvement. Interventions must assess stereotypes, emotional reactions, social distance, blame and behaviour separetly. === Structural and situational barriers === An exclusive focus on individual cognition can conceal structural stigma. Discrimination is embedded in policies, resource allocation, institutional routines and cultural representations, not only in personal beliefs (Link & Phelan, 2001). Teaching a growth mindset cannot compensate for inaccessible healthcare, discriminatory hiring or punitive. Indeed, an organisation may celebrate learning while leaving unequal systems untouched (Hoyt & Burnette, 2020). Much mindset and prejudice research uses Western, educated samples and US-specific eace relations. Cultural beliefs about agency, identity and responsibility may alter the meaning of growth. The conclusion is therefore conditional, effects depend on the stigma, audience, message and setting (Hoyt & Burnette, 2025) {{RoundBoxTop|theme=2}} '''Case study follow-up''' Emma's colleague learns that recovery is possible but is influenced by treatment access, social support, discrimination and life circumstances. Instead of assuming that Emma is permanately unrealiable or entirely responsible for her past difficulties, the colleague recognises both her capacity for growth and the barriers she has faced. {{RoundBoxBottom}} == How can mindsets inform stigma-reduction interventions? == An exclusive does not justify replacing all fixed messages with slogans about effort. It supports balanced growth mesages that combine realistic hope, effective pathways, contextual explanation and shared responsibility. Interventions should be designed around the mechanism relevant to a specific problem rather than treating "growth mindset" as a universal cure (Hoyt & Burnette, 2020, 2025). <big>'''Principle 1 - Separate changeability from person control'''</big> Messages should explicitly state that change can be possible without being immediate, easy or entirely voluntary. Recovery may require evidence-based treatment, stable housing, supportive relationships and repeated attempts. Likewise, prejudice can be learned and changed, but doing so requires reflection, corrective feedback and supportive social norms. This distinction reduces the likelihood that possibility will be translated into blame (Hoyt & Burnette, 2020). <big>'''Principle 2: Identify the appropriate target of change'''</big> The desired change may be in an observer's stereotype, an institution's practice or a social norm, not in the stigmatised person. In school's, students can learn that stereotypes are acquired interpretations that can be questioned. In workplaces, staff can develop inclusive skills while leaders change procedures that produce inequality. In healthcare, professionals can communicate hope about treatment while examining barriers, diagnostic bias and disrespectful care (Hoyt & Burnette, 2020; Levy and Dweck, 1999). <big>'''Principle 3 - Provide strategies and supportive conditions'''</big> A credible growth message describes how improvement can occur. Carr et al's (2012) and Neel and Shapiro's (2012) findings suggest that presenting prejudice as malleable may increase willingness to enter learning situations. An intervention could pair this message with structured intergroup cooperation, feedback, perspective taking and opportunities to practise responding to biased behaviour. Information alone is unlikely to be enough if the environment punishes mistakes or tolerates discrimination. '''<big>Principle 4 - Use narratives without creating exceptionalism</big>''' Stories of recovery can make development concrete but exceptional success stories may imply that continued struggle reflects failure. Narratives should represent varied pathways, setbacks and support, include lived experience, and avoid reducing people to inspirational examples (Hoyt & Burnette, 2020) <big>'''Principle 5 - Evaluate benefits and harms'''</big> Evaluation should measure more than immediate agreement with a growth statement. Relevant outcomes include essentialism, blame, anxiety, empathy and social distance, willingness for contact and discriminatory behaviour. Researchers should examine whether effects persist, whether they differ across groups and whether stigmatised people experience the intervention as respectful. Because opposing pathways can cancel each other out, mediation analyses can clarify whether reduced essentialism is accompanied by increased blame (Hoyt et al, 2017). Randomised designs can test casual effects, while longitudinal and field research can determine whether changes last and transfer into everyday conduct. Qualitative feedback is also valuable because fixed-choice scales may miss unintended meanings. Co-design with affected communities can identify language that researchers might otherwise assume is empowering. Finally, interventions should report null and adverse effects rather than treating them as failures to be hidden. === Practical applications === In a school program, students might learn that prejudiced beliefs are acquired and revisable, analyse how stereotypes are maintained, and practise responding to exclusion. The program should also establish clear anti-discrimination norms so that change is supported collectively. A workplace intervention could present inclusive communication as learnable while auditing recruitment and promotion outcomes. A mental health campaign could communicate that recovery is possible through varied pathways, while acknowledging that symptoms are not a choice and access to care is unequal. Across these settings, effective intervention follows a common logic, reduce the perceived permanance of negative stereptyes, increase confidence that constructive action matters, and prevent changeability from becoming a reaaon for blam. Growth is framed as supported development rather than a test of personal worth. ==Figures== [[File:Share with mental health or development disorder, COD.svg|alt=|thumb|140x140px|'''Figure 2 - The change of population of those with mental illness's over decades''']] *Three-quarters of Australian men and women survey respondents reported that they would be comfortable talking to their partner about mental health problems (figure 2).  Around one-sixth (16%) were not sure whether they would be comfortable, and a further ten per cent indicated they would not be comfortable [https://www.relationships.org.au/document/october-2015-mental-health-stigma/ (Mental Heath, 2026).] ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. ; ; ==Conclusion== Growth and fixed mindsets influence stigma because beliefs about change organise social judgement and motivation. Fixed mindsets can strengthen essentialism, encourage stable trait explanations and create pessimistic expectations that motive avoidance. Growth mindsets can weaken these processes by highlighting development and individual variation, creating hope, and increasing willingness to engage in intergroup learning. Research across stereotype formation and interracial interaction supports these mechanisms, although much of it relies on short term experiments and Western samples (Carr et al., 2012; Hoyt et al., 2024; Levy et al., 1998; Neel & Shapiro, 2012). Growth mindsets are not automatically compassionate. When changeability is interpreted as personal controllability, observers may blame people for not changing. This double-edged effect is particulary clear in weight-stigma research and may apply differently across conditions and cultures. The answer to the chapter's central question is therefore contextual, growth mindsets can reduce prejudice, but only when their casual and moral meanings are carefully designed (Burnette et al., 2017; Hoyt & Burnette, 2020; Hoyt et al., 2017). The most effective interventions communicate realistic possibilities for change while acknowledging stuctural barriers unequal resources and the need for social support. They direct responsibility toward observers and institutions as well as individuals, provide practical opportunities for learning, and evaluate both benefits and unintended harms. The central take-home message is not simply that people can change. It is that human development occurs in social conditions, and reducing stigma requires those conditions to change too. ==See also== * [[Motivation and emotion/Book/2018/Growth mindset and challenge|Growth mindset and challenge]] (Book chapter, 2018) * [[wikipedia:Mindset|Mindset]] (Wikipedia) * [[wikipedia:Psychological essentialism|Psychological essentialism]] (Wikipedia) * [[wikipedia:Social stigma|Social stigma]] (Wikipedia) ==References== {{Hanging indent|1= Burnette, J. L., O'Boyle, E. H., VanEpps, E. M., Pollack, J. M., & Finkel, E. J. (2013). Mind-sets matter: A meta-analytic review of implicit theories and self-regulation. ''Psychological Bulletin'', ''139''(3), 655–701. https://doi.org/10.1037/a0029531 Carr, P. B., Dweck, C. S., & Pauker, K. (2012). “Prejudiced” behavior without prejudice? Beliefs about the malleability of prejudice affect interracial interactions. ''Journal of Personality and Social Psychology'', ''103''(3), 452–471. https://doi.org/10.1037/a0028849 Chiu, C., Hong, Y., & Dweck, C. S. (1997). Lay dispositionism and implicit theories of personality. ''Journal of Personality and Social Psychology'', ''73''(1), 19–30. https://doi.org/10.1037/0022-3514.73.1.19 Hoyt, C. L., & Burnette, J. L. (2020). Growth mindset messaging in stigma-relevant contexts: Harnessing benefits without costs. ''Policy Insights from the Behavioral and Brain Sciences'', ''7''(2), 157–164. https://doi.org/10.1177/2372732220941216 Hoyt, C. L., & Burnette, J. L. (2025). How mindsets can mitigate or sustain prejudice. ''Current Directions in Psychological Science'', ''34''(2), 82–87. https://doi.org/10.1177/09637214241301290 Hoyt, C. L., Burnette, J. L., Auster-Gussman, L., Blodorn, A., & Major, B. (2017). The obesity stigma asymmetry model: The indirect and divergent effects of blame and changeability beliefs on antifat prejudice. ''Stigma and Health'', ''2''(1), 53–65. https://doi.org/10.1037/sah0000026 Hoyt, C. L., Rafferty, D., Earl, S., & Burnette, J. L. (2024). Growth mindsets of people can promote interracial contact intentions among White Americans via positive expectations and reduced anxiety. ''Group Processes & Intergroup Relations'', ''27''(4), 903–924. https://doi.org/10.1177/13684302231187262 Levy, S. R., & Dweck, C. S. (1999). The impact of children's static versus dynamic conceptions of people on stereotype formation. ''Child Development'', ''70''(5), 1163–1180. https://doi.org/10.1111/1467-8624.00085 Molden, D. C., & Dweck, C. S. (2006). Finding “meaning” in psychology: A lay theories approach to self-regulation, social perception, and social development. ''American Psychologist'', ''61''(3), 192–203. https://doi.org/10.1037/0003-066X.61.3.192 Neel, R., & Shapiro, J. R. (2012). Is racial bias malleable? Whites' lay theories of racial bias predict divergent strategies for interracial interactions. ''Journal of Personality and Social Psychology'', ''103''(1), 101–120. https://doi.org/10.1037/a0028237}} ==External links== * [https://implicit.harvard.edu/implicit/takeatest.html Project Implicit] (Harvard University) * [https://www.apa.org/topics/racism-bias-discrimination Racism, bias, and discrimination] (American Psychological Association) {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} * [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Mindset/Growth]] [[Category:Motivation and emotion/Book/Stigma]] dww34kf7tgdb1bygpaakgklziscf8zm 0 331095 2835625 2833546 2026-10-03T05:50:55Z JessJ117 3106965 2835625 wikitext text/x-wiki {{title|Alcohol use for emotion regulation: <br>Why and how do people use alcohol to regulate their emotions?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Alcohol-bar-party-cocktail (24218389642).jpg|right|thumb|150px|'''Figure 1'''. An alcoholic beverage on a bar bench.]] ;Consider this scenario It is Friday night; you've just completed a long week balancing university assessments, work and social engagements. You're feeling overwhelmed, stressed and mentally exhausted. You go out with some friends to celebrate the weekend and decide to have a few drinks. You start to notice your nervous system relaxing; you feel less tired, stop thinking about the stressful week you've had, and find your mind and body relaxing. What started as a few social drinks with friends turns into every night. You now find yourself dependent on alcohol to relax, drinking whenever you are overwhelmed or stressed. Now think: Are you actually reliant on the alcohol or the relief of overwhelming emotions that alcohol provides? {{RoundBoxBottom}} Further explanation...... {{RoundBoxTop|theme=3}} '''Focus questions''' * What is emotional regulation and how is it related to alcohol use? * Why do people use alcohol to regulate emotions? * What are the outcomes of excessive alcohol use? * How does alcohol influence the nervous system? {{RoundBoxBottom}} ==Defining emotional regulation and alcohol use== {{expand}} === What is emotion regulation? === Emotional regulation is a fundamental psychological process that allows individuals to manage and manipulate their own emotional responses. In alignment with research studied by Gross (1998), [https://static1.squarespace.com/static/55917f64e4b0cd3b4705b68c/t/5c47fda9758d46280be433e9/1548221901394/Gross+%281998%29.pdf emotional regulation] {{ic|Move link to References or External links}} is the process through which people influence the emotions they have, when they have them and the way in which they experience them. Process model of emotion regulation Gross (1998) foundational model of emotion regulation proposed that there are 5 main emotion regulation strategies including selection of the situation, modification of the situation, deployment of attention, change of cognitions, and modulation of responses. === What does regular emotional regulation look like and how does it change based on various situations? === === Define alcohol use (regular, exescesive etc) === == Motivations for using alcohol to regulate emotions == {{expand}} === What happens when alcohol enters the body === When alcohol first enters the body through oral consumption, it travels into the stomach and intestine and then passes into the bloodstream. Once in the bloodstream, alcohol distributes throughout the body's water until it reaches organs, specifically the brain. Once in the brain, alcohol interferes with the communication between neurotransmitter systems in the brain, which results in alterations in mood, behaviour, coordination, impulse control and memory (Goldman et al., 2025). Genetic factors '''How does alcohol effect emotional response?''' '''What are the potential motivations for drinking alcohol to regulate emotions?''' Individuals who experience difficulties in emotion regulation are the most susceptible to using alcohol as a method of self-regulation and will in term experience the most difficulty when trying to recover from an Alcohol use disorder (AUD) (Brown & Melas, 2024). Emotion dysregulation: (Gratz & Roemer, 2004) Emotion dysregulation is defined by the == Conquences of alcohol use for emotion regulation == '''What are the negative consequences of using alcohol for emotional regulation?''' '''What are the biological consequences of alcohol for emotional regulation?''' '''What are the short term and long-term consequences of alcohol use?''' Short term Long term Excessive and ongoing alcohol consumption can have extreme repercussions on the overall functioning of the entire nervous system, particularly the brain. These repercussions can include changes in emotions impaired perception and personality (Oscar-Berman et al., 2026). == Alternative strategies to manage emotion regulation == {{expand}} ==== Alternative strategies to manage emotion regulation ==== ==== Emotional regulation strategies ==== * Kozubal et al. (2023_ ==== Cognitive reappraisal ==== <quiz display="simple"> {True or False: Alcohol can be used to make someone feel relaxed for a short period of time, but it doesn’t solve the problems that is causing the stress: |type="()"} + True - False {Why might an individual decide to use alcohol for emotional regulation{{g}} |type="()"} - To increase their ability to problem solve. + To temporarily reduce or escape unpleasant emotions. - To improve their long-term emotional regulation - To permanently eliminate negative emotions </quiz> ==Conclusion== * Summarise alcohol use for emotional regulation (immediate reward for discomfort) * Effects of using alcohol for emotional regulation * Alternative strategies * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[Motivation and emotion/Book/2013/Alcohol and emotion|Alcohol and emotion]] (Book chapter, 2013) * [[Motivation and emotion/Book/2020/Cocaine and emotion regulation|Cocaine and emotion regulation]] (Book chapter, 2020) * [[Motivation and emotion/Book/2015/Alcohol addiction and emotion|Alcohol addiction and emotion]] (Book chapter, 2015) * [https://onlinelibrary.wiley.com/doi/full/10.1111/acer.12914 Emotion Regulation in Alcohol Dependence] {{ic|Move to References and cite}} * [https://www.frontiersin.org/journals/psychiatry/articles/10.3389/fpsyt.2019.01028/full Association Between Interoception and Emotion Regulation in Individuals With Alcohol Use Disorder] {{ic|Move to References and cite}} ==References== {{Hanging indent|1= Aurora, P., Klanecky, A. K., & Klanecky, A. K. (2016, February 23). ''Drinking motives mediate emotion regulation difficulties and problem drinking in college students''. The American Journal of Drug and Alcohol Abuse; Taylor & Francis. <nowiki>https://www.tandfonline.com/doi/full/10.3109/00952990.2015.1133633?utm_</nowiki> Brown, W. C., & Melas, M. T. (2024). A Narrative Review of Emotion Regulation Abilities, Strategy Use, and Their Relationship to Alcohol. ''Current Addiction Reports'', ''11''(2), 199–209. <nowiki>https://doi.org/10.1007/s40429-024-00553-3</nowiki> Goldman, M. R., Molina-Castro, M., Etkins, J. C., Koide, T. L., Ramchandani, V. A., Plawecki, M. H., Mennella, J. A., & Pepino, M. Y. (2025). Recent advances in alcohol metabolism: from the gut to the brain. Physiological Reviews, 105(4), 2501–2535. <nowiki>https://doi.org/10.1152/physrev.00053.2024</nowiki> Gratz, K. L., & Roemer, L. (2004). Multidimensional assessment of emotion regulation and dysregulation: Development, factor structure, and initial validation of the Difficulties in Emotion Regulation Scale. ''Journal of Psychopathology and Behavioral Assessment, 26''(1), 41–54. Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. <nowiki>https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271</nowiki> Kozubal, M., Szuster, A., & Wielgopolan, A. (2023). Emotional regulation strategies in daily life: the intensity of emotions and regulation choice. ''Frontiers in Psychology'', ''14''. <nowiki>https://doi.org/10.3389/fpsyg.2023.1218694</nowiki> Oscar-Berman, M., Shagrin, B., Evert, D. L., & Epstein, C. (2026). Impairments of Brain and Behavior: The Neurological Effects of Alcohol. ''Alcohol Health and Research World'', ''21''(1), 65. <nowiki>https://pmc.ncbi.nlm.nih.gov/articles/PMC6826797/</nowiki> }} ‌{{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Drugs/Alcohol]] [[Category:Motivation and emotion/Book/Emotion regulation]] edlpmt4szpjbnl8d8n0atk1h7ldpgrj 2835628 2835625 2026-10-03T05:58:20Z JessJ117 3106965 /* Overview */ 2835628 wikitext text/x-wiki {{title|Alcohol use for emotion regulation: <br>Why and how do people use alcohol to regulate their emotions?}} __TOC__ == Overview == {{RoundBoxTop|theme=4}} [[File:Alcohol-bar-party-cocktail (24218389642).jpg|right|thumb|150px|'''Figure 1'''. An alcoholic beverage on a bar bench.]] ;Consider this scenario It is Friday night; you've just completed a long week balancing university assessments, work and social engagements. You're feeling overwhelmed, stressed and mentally exhausted. You go out with some friends to celebrate the weekend and decide to have a few drinks. You start to notice your nervous system relaxing; you feel less tired, stop thinking about the stressful week you've had, and find your mind and body relaxing. What started as a few social drinks with friends turns into every night. You now find yourself dependent on alcohol to relax, drinking whenever you are overwhelmed or stressed. Now think: Are you actually reliant on the alcohol or the relief of overwhelming emotions that alcohol provides? {{RoundBoxBottom}} {{RoundBoxTop|theme=3}} '''Focus questions''' * What is emotional regulation and how is it related to alcohol use? * Why do people use alcohol to regulate emotions? * What are the outcomes of excessive alcohol use? * How does alcohol influence the nervous system? {{RoundBoxBottom}} ==Defining emotional regulation and alcohol use== {{expand}} === What is emotion regulation? === Emotional regulation is a fundamental psychological process that allows individuals to manage and manipulate their own emotional responses. In alignment with research studied by Gross (1998), [https://static1.squarespace.com/static/55917f64e4b0cd3b4705b68c/t/5c47fda9758d46280be433e9/1548221901394/Gross+%281998%29.pdf emotional regulation] {{ic|Move link to References or External links}} is the process through which people influence the emotions they have, when they have them and the way in which they experience them. Process model of emotion regulation Gross (1998) foundational model of emotion regulation proposed that there are 5 main emotion regulation strategies including selection of the situation, modification of the situation, deployment of attention, change of cognitions, and modulation of responses. === What does regular emotional regulation look like and how does it change based on various situations? === === Define alcohol use (regular, exescesive etc) === == Motivations for using alcohol to regulate emotions == {{expand}} === What happens when alcohol enters the body === When alcohol first enters the body through oral consumption, it travels into the stomach and intestine and then passes into the bloodstream. Once in the bloodstream, alcohol distributes throughout the body's water until it reaches organs, specifically the brain. Once in the brain, alcohol interferes with the communication between neurotransmitter systems in the brain, which results in alterations in mood, behaviour, coordination, impulse control and memory (Goldman et al., 2025). Genetic factors '''How does alcohol effect emotional response?''' '''What are the potential motivations for drinking alcohol to regulate emotions?''' Individuals who experience difficulties in emotion regulation are the most susceptible to using alcohol as a method of self-regulation and will in term experience the most difficulty when trying to recover from an Alcohol use disorder (AUD) (Brown & Melas, 2024). Emotion dysregulation: (Gratz & Roemer, 2004) Emotion dysregulation is defined by the == Conquences of alcohol use for emotion regulation == '''What are the negative consequences of using alcohol for emotional regulation?''' '''What are the biological consequences of alcohol for emotional regulation?''' '''What are the short term and long-term consequences of alcohol use?''' Short term Long term Excessive and ongoing alcohol consumption can have extreme repercussions on the overall functioning of the entire nervous system, particularly the brain. These repercussions can include changes in emotions impaired perception and personality (Oscar-Berman et al., 2026). == Alternative strategies to manage emotion regulation == {{expand}} ==== Alternative strategies to manage emotion regulation ==== ==== Emotional regulation strategies ==== * Kozubal et al. (2023_ ==== Cognitive reappraisal ==== <quiz display="simple"> {True or False: Alcohol can be used to make someone feel relaxed for a short period of time, but it doesn’t solve the problems that is causing the stress: |type="()"} + True - False {Why might an individual decide to use alcohol for emotional regulation{{g}} |type="()"} - To increase their ability to problem solve. + To temporarily reduce or escape unpleasant emotions. - To improve their long-term emotional regulation - To permanently eliminate negative emotions </quiz> ==Conclusion== * Summarise alcohol use for emotional regulation (immediate reward for discomfort) * Effects of using alcohol for emotional regulation * Alternative strategies * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[Motivation and emotion/Book/2013/Alcohol and emotion|Alcohol and emotion]] (Book chapter, 2013) * [[Motivation and emotion/Book/2020/Cocaine and emotion regulation|Cocaine and emotion regulation]] (Book chapter, 2020) * [[Motivation and emotion/Book/2015/Alcohol addiction and emotion|Alcohol addiction and emotion]] (Book chapter, 2015) * [https://onlinelibrary.wiley.com/doi/full/10.1111/acer.12914 Emotion Regulation in Alcohol Dependence] {{ic|Move to References and cite}} * [https://www.frontiersin.org/journals/psychiatry/articles/10.3389/fpsyt.2019.01028/full Association Between Interoception and Emotion Regulation in Individuals With Alcohol Use Disorder] {{ic|Move to References and cite}} ==References== {{Hanging indent|1= Aurora, P., Klanecky, A. K., & Klanecky, A. K. (2016, February 23). ''Drinking motives mediate emotion regulation difficulties and problem drinking in college students''. The American Journal of Drug and Alcohol Abuse; Taylor & Francis. <nowiki>https://www.tandfonline.com/doi/full/10.3109/00952990.2015.1133633?utm_</nowiki> Brown, W. C., & Melas, M. T. (2024). A Narrative Review of Emotion Regulation Abilities, Strategy Use, and Their Relationship to Alcohol. ''Current Addiction Reports'', ''11''(2), 199–209. <nowiki>https://doi.org/10.1007/s40429-024-00553-3</nowiki> Goldman, M. R., Molina-Castro, M., Etkins, J. C., Koide, T. L., Ramchandani, V. A., Plawecki, M. H., Mennella, J. A., & Pepino, M. Y. (2025). Recent advances in alcohol metabolism: from the gut to the brain. Physiological Reviews, 105(4), 2501–2535. <nowiki>https://doi.org/10.1152/physrev.00053.2024</nowiki> Gratz, K. L., & Roemer, L. (2004). Multidimensional assessment of emotion regulation and dysregulation: Development, factor structure, and initial validation of the Difficulties in Emotion Regulation Scale. ''Journal of Psychopathology and Behavioral Assessment, 26''(1), 41–54. Gross, J. J. (1998). The emerging field of emotion regulation: An integrative review. ''Review of General Psychology'', ''2''(3), 271–299. <nowiki>https://doi.org/https://doi.org/10.1037/1089-2680.2.3.271</nowiki> Kozubal, M., Szuster, A., & Wielgopolan, A. (2023). Emotional regulation strategies in daily life: the intensity of emotions and regulation choice. ''Frontiers in Psychology'', ''14''. <nowiki>https://doi.org/10.3389/fpsyg.2023.1218694</nowiki> Oscar-Berman, M., Shagrin, B., Evert, D. L., & Epstein, C. (2026). Impairments of Brain and Behavior: The Neurological Effects of Alcohol. ''Alcohol Health and Research World'', ''21''(1), 65. <nowiki>https://pmc.ncbi.nlm.nih.gov/articles/PMC6826797/</nowiki> }} ‌{{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Drugs/Alcohol]] [[Category:Motivation and emotion/Book/Emotion regulation]] idmlmzt32xdq5o13pvn1wagasg88pde 0 331100 2835669 2835427 2026-10-03T10:48:05Z U3239236 3106753 fixed table 1 2835669 wikitext text/x-wiki {{title|Emotional intelligence:<br>How does emotional intelligence affect emotional wellbeing?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Sophie emotional regulation scenario.png|thumb|'''Figure 1.''' A university student pausing to regulate her emotions after receiving a disappointing grade, generated with ai ]] '''Scenario''' Imagine Sophie, a university student managing several assessments and conflict with a close friend. After receiving a disappointing mark, she feels frustrated, embarrassed, and overwhelmed. Rather than reacting immediately, she recognises her emotions, takes time to regulate them, and later talks calmly with her friend. Another student in the same situation might become overwhelmed or react impulsively. Why might people respond differently to similar emotional situations? One explanation involves emotional intelligence, the ability to perceive, understand, use, and regulate emotions. EI may influence how people interpret and respond to emotional experiences and, consequently, their emotional wellbeing. Emotional wellbeing involves more than experiencing positive emotions or avoiding negative ones; it also concerns how people experience and manage emotions. Understanding the relationship between EI and wellbeing can help explain which emotional abilities may support wellbeing, which components are particularly important, and the psychological processes underlying this relationship. {{RoundBoxBottom}} {{RoundBoxTop}} ; Focus questions # What are emotional intelligence and emotional wellbeing? # What is the relationship between emotional intelligence and emotional wellbeing? #How does emotional intelligence influence emotional wellbeing? #Which components of emotional intelligence are particularly important for emotional wellbeing? # How can emotional intelligence be developed to support emotional wellbeing? {{RoundBoxBottom}} == What are emotional intelligence and emotional wellbeing? == Emotional intelligence can play an important role in how people recognise, understand, and regulate their emotions. These emotional abilities may influence how people cope with difficult experiences and support their overall emotional wellbeing. However, EI is a complex construct, and different emotional abilities may contribute to wellbeing in different ways. Before examining this relationship, it is important to understand what psychologists mean by emotional intelligence and emotional wellbeing. One influential approach to emotional intelligence is Mayer and Salovey's ability model, which describes emotional intelligence through four related abilities: perceiving emotions, using emotions to support thinking, understanding emotions, and managing emotions see Figure 1. An important practical question is whether emotional abilities can be developed and whether improving them leads to better wellbeing.[[File:Model of emotional intelligence.png|thumb|'''Figure 1.''' ''Mayer and Salovey's Emotional Intelligence model'' |359x359px]] === Defining emotional wellbeing === Emotional wellbeing refers to people's emotional experiences and how they evaluate their lives. Psychological research often distinguishes between hedonic and eudaimonic approaches to wellbeing. Hedonic wellbeing focuses on experiences such as pleasure, enjoyment, and comfort, whereas eudaimonic wellbeing emphasises personal growth, meaning, and authenticity (Huta & Waterman, 2014). Both approaches are important for understanding wellbeing, although researchers differ in how they define and measure these concepts. Importantly, emotional wellbeing does not mean feeling happy or positive all the time. Negative emotions such as sadness, anger, anxiety, and disappointment are normal responses to difficult experiences. Chen et al. (2023) explain that adaptive stress responses can help people cope with stressful situations or adjust to challenges that cannot be resolved, whereas maladaptive responses can increase the risk of stress-related difficulties. This highlights why wellbeing involves more than simply experiencing positive emotions; how people respond to difficult experiences is also important. Another approach to understanding wellbeing is Seligman's PERMA model, which identifies five elements of flourishing: positive emotion, engagement, relationships, meaning, and accomplishment (Al-Hendawi et al., 2024). Unlike approaches that focus mainly on emotional experiences, PERMA considers several aspects of a fulfilling life see Figure 2.[[File:ChatGPT Image Sep 26, 2026, 01 48 17 PM.png|thumb|335x335px|'''Figure 2.''' ''Seligman's PERMA model of wellbeing, consisting of positive emotion, engagement, relationships, meaning, and accomplishment. (AI generated)'' ]]<quiz display="simple"> Quiz 1. Which statement best describes emotional intelligence? |type="()"} + The ability to perceive, understand, use, and manage emotions. - The ability to avoid experiencing negative emotions. - The ability to remain happy in all situations. - The ability to control the emotions of other people. } </quiz> <quiz display="simple"> Quiz 2. What does emotion regulation involve? |type="()"} + Managing how we experience and respond to emotions. - Avoiding all negative emotions. - Staying positive in stressful situations. - Ignoring emotions until they disappear. } </quiz> === Trait emotional intelligence === Trait EI focuses on how people perceive their own emotional abilities and tendencies rather than how they perform on an objective test. It is usually measured through self-report questionnaires that assess how people typically recognise, understand, and manage emotions. In contrast, performance-based tests assess emotional abilities through emotion-related tasks. Chapman (2005) explains that trait EI is more closely related to personality, whereas ability EI is more closely associated with cognitive abilities. Performance-based measures, such as the Multifactor Emotional Intelligence Scale and Mayer-Salovey-Caruso Emotional Intelligence Test, assess responses using scoring criteria. However, determining a correct emotional response can be difficult because emotional situations may be interpreted differently. Expert scoring relies on selected experts, while consensus scoring reflects the responses of a reference group (Chapman, 2005). Trait EI avoids this scoring issue by focusing on people's perceptions of their emotional abilities. Common self-report measures include the Trait Meta-Mood Scale and the 33-item Schutte Self-Report Inventory of Emotional Intelligence (Chapman, 2005). However, people's perceptions of their emotional abilities may not always reflect their actual performance. This distinction is important when examining emotional wellbeing. Someone may believe they manage emotions effectively even when their behaviour suggests otherwise. Trait EI and wellbeing are also often measured using self-report questionnaires, which may influence the strength of the relationship observed between them (Sánchez-Álvarez et al., 2016). == What is the relationship between emotional intelligence and emotional wellbeing? == === Emotional intelligence and positive wellbeing === Research generally suggests that people with higher EI report greater subjective wellbeing. A meta-analysis of 25 studies involving more than 8,500 participants found a moderate positive association between EI and subjective wellbeing (''r'' = .32; Sánchez-Álvarez et al., 2016). This indicates that higher EI tends to be associated with greater subjective wellbeing, although the relationship is moderate rather than strong. Importantly, the strength of the relationship depended on how EI was measured. The association was stronger for self-report measures that incorporated emotional and personality-related characteristics (''r'' = .38) than for performance-based measures (''r'' = .22). EI also showed a somewhat stronger relationship with life satisfaction (''r'' = .35) than with emotional experiences (''r'' = .29). These findings demonstrate why the way EI and wellbeing are defined and measured matters when interpreting their relationship (Sánchez-Álvarez et al., 2016). However, these findings do not demonstrate that EI causes greater happiness or life satisfaction. People who already experience greater wellbeing may find it easier to perceive or manage emotions, while other characteristics, including personality, could potentially contribute to both EI and wellbeing. === Emotional intelligence and negative wellbeing === EI has also been examined in relation to psychological distress and other negative outcomes. Martins et al. (2010) found that higher EI was associated with better health outcomes, including aspects of mental health. Research has also examined relationships between emotional abilities, emotion regulation, and symptoms of depression (Fernández-Berrocal & Extremera, 2016). Importantly, higher EI does not mean that someone will never experience negative emotions. Instead, emotional abilities may influence how people understand and respond to difficult emotions when they occur. Sophie's emotional abilities, for example, do not prevent her from feeling disappointed about her grade. They may instead help her recognise the emotion and choose how to respond to it. === Trait and ability emotional intelligence === Trait EI tends to show stronger associations with wellbeing than ability EI, which is typically assessed using performance-based measures. One possible explanation is that trait EI and wellbeing are often both assessed using self-report questionnaires. Similar measurement methods can contribute to stronger observed relationships between constructs. Self-report measures also assess people's perceptions of their emotional abilities, which may not always correspond with their performance on emotional ability tasks. This makes the way EI is measured particularly important when interpreting its relationship with wellbeing === '''What does the evidence tell us?''' === Overall, research suggests that higher EI is linked to greater subjective wellbeing and lower psychological distress. However, the strength of this relationship varies depending on how EI is defined and measured (Martins et al., 2010; Sánchez-Álvarez et al., 2016). However, an association between EI and wellbeing does not demonstrate that EI directly causes better wellbeing. Much of this research is correlational, meaning the direction of the relationship cannot necessarily be established and other factors may contribute to both constructs. Differences between EI models and measurement approaches further complicate interpretation. Therefore, the evidence supports a relationship between EI and wellbeing, but stronger causal conclusions require research capable of examining whether changes in EI actually produce changes in wellbeing. Finding an association also does not explain how EI and wellbeing are connected. Emotional awareness, emotional understanding, emotion regulation, coping with stress, and social relationships may provide possible pathways through which EI is associated with emotional wellbeing. == How does emotional intelligence influence emotional wellbeing? == === Emotional awareness and perception === The relationship between emotional intelligence and wellbeing may be explained by several psychological processes. Emotional intelligence may help people recognise what they are feeling, understand why those emotions have occurred, regulate difficult emotional responses, and cope more effectively with stressful situations. It may also support communication and relationships with others. Rather than working separately, these processes are likely to interact and may help explain why people with higher emotional intelligence often report greater wellbeing. === Emotional awareness and perception === Recognising emotions is an important first step in responding to them effectively. Being able to identify whether a person is feeling anxious, angry, disappointed, or overwhelmed can provide useful information about what is happening and what they may need to do next. For example, recognising feelings of anxiety before an assessment may help a student identify the source of their stress and decide whether they need to prepare further, seek support, or use a strategy to manage their anxiety. Difficulty recognising emotions may make it harder to respond appropriately. A person who interprets anxiety as anger, for example, may react to a situation differently than someone who accurately understands what they are experiencing. Emotional awareness alone does not guarantee good emotional wellbeing, but it may provide the information needed for later processes such as emotional understanding, regulation, and coping (Lane & Smith, 2021). {{RoundBoxTop|theme=3}}💡 Tip When you experience a strong emotion, try identifying exactly what you are feeling rather than simply describing it as “good” or “bad.” For example, you might recognise that you are disappointed rather than embarrassed, or anxious rather than angry. This ability to identify emotions more precisely, known as emotional granularity, can help build emotional awareness and guide how you respond.{{RoundBoxBottom}} === Understanding emotions === Emotional intelligence also involves understanding why emotions occur and how they may change over time. Emotional experiences are not always simple, and people can experience several emotions at once. Someone receiving critical feedback, for example, might simultaneously feel disappointed, embarrassed, and motivated to improve. Understanding the causes and possible consequences of these emotions may help people choose more appropriate responses. Instead of immediately reacting to disappointment, a person who understands where the feeling comes from may be better able to consider the situation before deciding what to do (Tzieropoulos et al., 2011). Emotional understanding may therefore contribute to wellbeing indirectly by supporting more effective emotion regulation and coping. === Emotion regulation === Emotion regulation refers to how people influence their emotional experiences and responses. Rather than eliminating negative emotions, regulation can involve changing how a situation is interpreted or choosing how to respond. Different strategies may have different consequences for wellbeing. Cognitive reappraisal changes the interpretation of a situation to alter its emotional impact, however suppression reduces emotional expression without necessarily changing the underlying experience. Emotional intelligence may help people select appropriate regulation strategies, potentially explaining its relationship with wellbeing (MacCann et al., 2026). However, further evidence is needed to determine whether emotion regulation actually mediates the relationship between emotional intelligence and wellbeing. === Coping with stress === Emotional intelligence may also influence how people respond to stressful experiences. Stress can produce difficult emotional responses such as tension, worry, and feelings of being overwhelmed. Chen et al. (2023) distinguish between adaptive responses that help people cope with or adjust to stressors and maladaptive responses that may interfere with functioning and increase the risk of stress-related difficulties. Emotional skills may be useful during this process because recognising and understanding an emotional response can help a person decide how to cope with the situation. Effective coping does not necessarily remove the stressor, particularly when a situation cannot be changed. Instead, people may need to adjust their response to it. Emotion regulation has also been identified as a resilience-related factor, further demonstrating the connection between emotional processes and responses to stress. === Social relationships and support === Emotional intelligence may also contribute to wellbeing through relationships with other people. Recognising another person's emotions can make it easier to understand their perspective, communicate appropriately, and respond to interpersonal difficulties. Managing one's own emotional responses may also be important during disagreements or emotionally challenging conversations. These abilities could support stronger relationships and access to social support, providing another possible pathway between emotional intelligence and wellbeing. However, being able to recognise another person's emotions does not automatically lead to supportive or prosocial behaviour (Hidalgo-Fuentes et al., 2024). Emotional abilities describe how effectively emotional information is processed, rather than whether that information will always be used positively. '''Figure 3.''' ''Possible pathways through which emotional intelligence may support emotional wellbeing.'' Emotional intelligence may contribute to emotional wellbeing through several interacting processes, including emotional awareness, emotional understanding, emotion regulation, coping with stress, and social functioning. These processes may influence one another rather than occurring in a simple step-by-step sequence. == Which components of emotional intelligence are particularly important for emotional wellbeing? == Although emotional intelligence is often discussed as a single construct, Mayer and Salovey's ability model proposes several distinct emotional abilities. These components may not contribute equally to emotional wellbeing. Comparing emotion perception, using emotions, emotional understanding, and emotion management can therefore provide a clearer picture of which aspects of emotional intelligence may be particularly relevant to wellbeing. === Emotion perception === Emotion perception involves accurately recognising emotions in oneself and other people. This ability may contribute to wellbeing by helping individuals identify their emotional state and recognise emotional information in social situations. However, recognising an emotion does not necessarily mean that a person knows how to respond to it. Someone may accurately recognise that they are angry, for example, while still responding impulsively. Emotion perception may therefore provide an important foundation for emotional functioning without being sufficient on its own to support wellbeing (Carfagno, 2020). === '''Using emotions''' === Using emotions involves drawing on emotional information to support thinking, attention, judgement, and decision-making. Different emotional states can influence what people notice and how they approach a problem. Being able to use this information may therefore help individuals adapt their thinking to different situations. However, the relevance of this component to emotional wellbeing needs to be established through research rather than assumed (Blasco-Belled et al., 2019). Compared with emotion management, for example, using emotions may have a less direct connection with how people cope with distress or regulate difficult emotional experiences. === Emotion understanding === Emotion understanding involves recognising the causes of emotions, distinguishing between related emotional states, and understanding how emotions can change or combine. This ability may help people make sense of complicated emotional experiences and anticipate how feelings might develop. Understanding an emotion may also make it easier to choose an appropriate coping or regulation strategy. However, as with emotion perception, understanding an emotion does not guarantee that it will be managed effectively. Research comparing the different branches of emotional intelligence is therefore needed to determine whether emotional understanding has an independent relationship with wellbeing or primarily contributes through other emotional processes (Antonopoulou, 2024). === Emotion regulation and management === Emotion management may have a particularly direct connection with emotional wellbeing because it concerns how people respond to and regulate emotional experiences. Effective regulation could help people manage distress, maintain positive emotional functioning, and recover from stressful experiences. This does not mean eliminating negative emotions, but responding to them in ways that are appropriate to the situation. However, emotion management should not automatically be considered the most important component of emotional intelligence. Its relative importance needs to be established by studies that compare it directly with perception, use, and understanding of emotions. Differences in how emotional intelligence and wellbeing are measured may also affect which component appears most strongly related to wellbeing (Blasco-Belled et al., 2019). === Comparing the components === Overall, the four emotional abilities may contribute to wellbeing in different but connected ways. Perception provides information about emotional states, understanding helps make sense of that information, using emotions may support thinking and decision-making, and management concerns how emotional experiences are handled. Rather than functioning independently, these abilities may work together. Determining whether one component is particularly important requires direct comparisons across the four branches. Evidence that emotion management is associated with wellbeing would not, by itself, demonstrate that it is more important than the other components. Differences between ability-based and self-report measures should also be considered when interpreting these findings. <quiz display="simple"> Quiz 3. Which statement best describes how the components of emotional intelligence relate to wellbeing? |type="()"} + Different EI components may contribute to wellbeing in different but connected ways. - Emotion management is always the most important component for wellbeing. - Emotion perception alone is enough to support emotional wellbeing. - All EI components affect wellbeing in exactly the same way. } </quiz> <quiz display="simple"> Quiz 4. Why might recognising an emotion not be enough to support wellbeing? |type="()"} + A person may recognise an emotion but still respond to it poorly. - Recognising emotions prevents people from regulating them. - Emotion perception only involves recognising other people's emotions. - Recognising emotions automatically removes negative feelings. } </quiz> '''Table 1.''' ''Emotional intelligence components and wellbeing'' {| class="wikitable" !EI component !What it involves !Relevance to wellbeing |- |'''Emotion perception''' |Recognising emotions |Supports emotional awareness |- |'''Using emotions''' |Using emotions to guide thinking |Supports attention and judgement |- |'''Emotion understanding''' |Understanding causes and changes in emotions |Supports understanding and responding |- |'''Emotion management''' |Regulating emotional responses |Supports coping and emotional functioning |} == How can emotional intelligence be developed to support emotional wellbeing? == If emotional intelligence contributes to wellbeing, an important practical question is whether these emotional abilities can be developed. Emotional intelligence interventions have attempted to improve skills such as emotional awareness, understanding emotions, emotion regulation, and interpersonal communication. However, evidence that EI can be improved does not necessarily mean that increasing EI will automatically improve emotional wellbeing. === Can emotional intelligence be developed? === Emotional intelligence is not necessarily a completely fixed characteristic. Training programs have attempted to develop emotional abilities through education, practice, feedback, and exercises focused on recognising and managing emotions. However, the effectiveness of training may depend on how emotional intelligence is conceptualised and measured. Changes in self-reported EI, for example, may indicate that people feel more confident in their emotional abilities without necessarily demonstrating equivalent improvements on performance-based measures. Research evaluating EI interventions is therefore important for determining whether emotional abilities can genuinely be improved and whether any improvements continue after training has ended (Hodzic et al., 2017). === Emotional intelligence interventions === EI interventions may target several skills rather than emotional intelligence as a single ability. Activities can focus on recognising emotional cues, developing emotional vocabulary, understanding emotional triggers, considering other people's perspectives, practising emotion-regulation strategies, and improving emotional communication. These approaches are particularly relevant to emotional wellbeing if improvements in emotional skills help people respond more effectively to stress and difficult emotional experiences. However, intervention studies need to examine more than whether EI scores increase. They should also assess whether changes are accompanied by meaningful improvements in wellbeing. === Do improvements in emotional intelligence improve wellbeing? === An important distinction is whether EI training improves emotional intelligence itself and whether those improvements subsequently lead to better wellbeing. An intervention could increase participants' knowledge about emotions without meaningfully changing their happiness, life satisfaction, stress, or emotional functioning. Evidence that an intervention improves both EI and wellbeing would provide stronger support for a possible causal relationship than correlational studies showing that the two are associated. Even then, researchers need to consider whether improvements are maintained over time and whether changes in wellbeing are actually explained by increased emotional intelligence (Hodzic et al., 2018; Nadler et al., 2020). === Limitations and practical implications === Although EI training may have practical value, its potential benefits should not be overstated. Emotional wellbeing is influenced by many personal, social, and environmental factors, and improving emotional skills cannot remove every source of stress or psychological difficulty. Differences between EI models, measurement methods, intervention designs, and participant groups can also make findings difficult to compare. ==Conclusion== Emotional wellbeing involves more than simply feeling happy or avoiding negative emotions. It also involves how people understand, respond to, and manage their emotional experiences. Overall, emotional intelligence may support emotional wellbeing by helping people recognise, understand, and regulate emotions and respond more effectively to stressful and interpersonal experiences. Research generally shows that higher EI is associated with better indicators of wellbeing, although the strength of this relationship varies depending on how EI and wellbeing are defined and measured. Importantly, emotional intelligence should not be viewed as a guarantee or universal cause of emotional wellbeing. Wellbeing is influenced by many factors, and emotionally intelligent people will still experience stress, disappointment, conflict, and other difficult emotions. Sophie's situation demonstrates this distinction. Her emotional abilities do not prevent her from feeling disappointed or overwhelmed; instead, they may help her recognise what she is experiencing, understand why she feels that way, regulate her response, and choose behaviours that better support her wellbeing ==See also Internal Links == * [[Motivation and emotion/Textbook/Emotion/Emotion management|Managing emotions]] * [https://en.wikipedia.org/wiki/Subjective_well-being?wprov=srpw1_1 Subjective wellbeing] (Book chapter, 2026) * [[Motivation and emotion/Book/2011/Emotional intelligence|Emotional intelligence]] (Book chapter, 2011) * [[wikibooks:Foundations_of_Education_and_Instructional_Assessment/Effective_Teaching/Intelligence#Introduction_to_Emotional_Intelligence|How are all children smart?]] (Wikibooks) ==References== {{Hanging indent|1= Antonopoulou, H. (2024, April 11). The value of emotional intelligence: Self-awareness, self-regulation, motivation, and empathy as key. ''Technium Education and Humanities'' . ResearchGate. https://www.researchgate.net/publication/379764627_The_Value_of_Emotional_Intelligence_Self-Awareness_Self-Regulation_Motivation_and_Empathy_as_Key_Components Al-Hendawi, M., Alodat, A., Al-Zoubi, S., & Bulut, S. (2024). A PERMA model approach to well-being: A psychometric properties study. ''BMC Psychology'', 12(1). https://doi.org/10.1186/s40359-024-01909-0 Blasco-Belled, A., Rogoza, R., Torrelles-Nadal, C., & Alsinet, C. (2019). Emotional intelligence structure and its relationship with life satisfaction and happiness: New findings from the bifactor model. ''Journal of Happiness Studies'', 21(6), 2031–2049. https://doi.org/10.1007/s10902-019-00167-x Chapman, B. P. (2005). Emotional intelligence at mid life: A cross sectional investigation of structural variance, social correlates, and relationship to established personality and ability taxonomies. Proquest.Com. https://www.proquest.com/psychology/docview/305400900/8D98674B36744231PQ/2?accountid=28889&sourcetype=Dissertations%20&%20Theses Chen, C., Kotozaki , Y., Okubo , R., & Nakagawa , S. (2023, July 13). Editorial: New insights into stress coping and resilience . Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC10374303/ Hidalgo-Fuentes, S., Martínez-Álvarez, I., Sospedra-Baeza, M. J., Martí-Vilar, M., Merino-Soto, C., & Toledano-Toledano, F. (2024). Emotional intelligence and perceived social support: Its relationship with subjective well-being. ''Healthcare'' (Basel, Switzerland), 12(6), 634. https://doi.org/10.3390/healthcare12060634 Hodzic, S., Scharfen, J., Ripoll, P., Holling, H., & Zenasni, F. (2017). How efficient are emotional intelligence trainings: A meta-analysis. ''Emotion Review'', 10(2), 138–148. https://doi.org/10.1177/1754073917708613 Huta, V., & Waterman, A. S. (2014). Eudaimonia and its distinction from hedonia: Developing a classification and terminology for understanding conceptual and operational definitions. ''Journal of Happiness Studies'', 15(6), 1425–1456. https://doi.org/10.1007/s10902-013-9485-0 Lane, R. D., & Smith, R. (2021). Levels of emotional awareness: Theory and measurement of a socio-emotional skill. ''Journal of Intelligence'', 9(3), 42. https://doi.org/10.3390/jintelligence9030042 Martins, A., Ramalho, N., & Morin, E. (2010). A comprehensive meta-analysis of the relationship between emotional intelligence and health. ''Personality and Individual Differences'', 49(6), 554–564. https://doi.org/10.1016/j.paid.2010.05.029 MacCann , C., Double , K. S., & E Clarke , I. (2026). Lower avoidant coping mediates the relationship of emotional intelligence with well-being and ill-being. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC9396212/ Nadler, R., Carswell , J., & Minda, J. P. (2020, February). Online mindfulness training increases well-being, trait emotional intelligence, and workplace competency ratings: A randomized waitlist-controlled trial. Canberra.Edu.Au. https://pmc-ncbi-nlm-nih-gov.ezproxy.canberra.edu.au/articles/PMC7048000/ Sánchez-Álvarez, N., Extremera, N., & Fernández-Berrocal, P. (2015). The relation between emotional intelligence and subjective well-being: A meta-analytic investigation. ''The Journal of Positive Psychology'', 11(3), 276–285. https://doi.org/10.1080/17439760.2015.1058968 Tzieropoulos, H., Peralta, R. G. de, Bossaerts, P., & Andino, S. L. G. (2011). The impact of disappointment in decision making: Inter-individual differences and electrical neuroimaging. ''Frontiers in Human Neuroscience'', 4(235). https://doi.org/10.3389/fnhum.2010.00235 }} ==External links== * [https://www.ted.com/talks/maximilian_park_emotional_intelligence_from_a_teenage_perspective Emotional Intelligence From a Teenage Perspective] * [https://rickhanson.com/being-well-podcast-emotional-intelligence-improving-self-awareness-self-regulation-and-empathy-2/ Being Well Podcast: Emotional Intelligence: Improving Self-Awareness, Self-Regulation, and Empathy] * [https://www.youtube.com/watch?v=4KbSRXP0wik Emotion, Stress, and Health: Crash Course Psychology #26] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Emotional intelligence]] [[Category:Motivation and emotion/Book/Well-being]] truwxttshomonx8io4595dxylu3mza8 0 331103 2835624 2834946 2026-10-03T05:50:31Z U3247927 3005952 2835624 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way - belief that love can overcome all obstacles''' * '''One and only - belief that we have "soulmates"''' * '''Idealisation - belief that a "true love" will nearly be perfect''' * '''Love at first sight - belief when you meet the right person, you will know''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment == Psychological mechanisms of media influence == '''Social exchange theory (SET)''' In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives. Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships. === Romantic scripts and schemas === === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: Soulmates (Wikipedia) Love (Book chapter, 2011) Parasocial interaction (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] lttynvadar0tv8pz2l6s831wbxofbuw 2835627 2835624 2026-10-03T05:55:03Z U3247927 3005952 /* See also */ 2835627 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. It can be thought as triangle vertex with intimacy, desire and decision/ commitment, the intensity of the three points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way - belief that love can overcome all obstacles''' * '''One and only - belief that we have "soulmates"''' * '''Idealisation - belief that a "true love" will nearly be perfect''' * '''Love at first sight - belief when you meet the right person, you will know''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment == Psychological mechanisms of media influence == '''Social exchange theory (SET)''' In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives. Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships. === Romantic scripts and schemas === === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[wikipedia:Soulmate|Soulmate]] (Wikipedia) * [[Motivation and emotion/Book/2011/Love|Love]] (Book chapter, 2011) * [[wikipedia:Parasocial_interaction|Parasocial interaction]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] bnc0ag1rpsyv90pz0ft0q51u6u7nozb 2835629 2835627 2026-10-03T05:58:50Z U3247927 3005952 love section 2835629 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. According to Sternberg's triangular theory of love, romantic love consists of three core components: intimacy, desire and decision/ commitment, the intensity of the points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way - belief that love can overcome all obstacles''' * '''One and only - belief that we have "soulmates"''' * '''Idealisation - belief that a "true love" will nearly be perfect''' * '''Love at first sight - belief when you meet the right person, you will know''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment == Psychological mechanisms of media influence == '''Social exchange theory (SET)''' In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives. Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships. === Romantic scripts and schemas === === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[wikipedia:Soulmate|Soulmate]] (Wikipedia) * [[Motivation and emotion/Book/2011/Love|Love]] (Book chapter, 2011) * [[wikipedia:Parasocial_interaction|Parasocial interaction]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] 7gmggx1n01fvaom6tnuk833khlp9qq4 2835634 2835629 2026-10-03T06:17:22Z U3247927 3005952 /* External links update*/ 2835634 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. According to Sternberg's triangular theory of love, romantic love consists of three core components: intimacy, desire and decision/ commitment, the intensity of the points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way - belief that love can overcome all obstacles''' * '''One and only - belief that we have "soulmates"''' * '''Idealisation - belief that a "true love" will nearly be perfect''' * '''Love at first sight - belief when you meet the right person, you will know''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are influenced by a range of social and cultural factors; however, these beliefs do not develop in isolation. Entertainment media play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Studies examining romantic comedy exposure suggest that romantic entertainment may contribute to the reinforcement of romantic ideals. Hefner and Wilson (2013) found that individuals who watched romantic comedies with the intention of learning about relationships were more likely to endorse romantic beliefs, particularly the idealisation of romantic partners. Furthermore, romantic comedies frequently depict themes associated with romantic ideals, including love at first sight, soulmates, idealised partners, and the belief that love can overcome obstacles. These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment == Psychological mechanisms of media influence == '''Social exchange theory (SET)''' In every interpersonal relation, a social exchange takes place where-by people tend to balance their emotional investment in the relationship by constantly comparing their current relationship to their perceptions of what they deserve. Rewards - love, companionship, emotional, acts of affection are considered to be what an individual enjoys most in the relationships. Social exchange Theory proposes that people evaluate romantic relationships by comparing perceived rewards with the perceived costs - conflict, compromise or missed alternative relationships. An individual satisfaction with his or her her relationship is influenced by whether the benefits outweigh the costs, as well as how closely the relationship matches the individuals expectations and perceived alternatives. Within the context of romantic entertainment, '''SET''' helps explain how heavy consumption of media may shape perceptions of a romantic relationships. === Romantic scripts and schemas === === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External linksEven Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[wikipedia:Soulmate|Soulmate]] (Wikipedia) * [[Motivation and emotion/Book/2011/Love|Love]] (Book chapter, 2011) * [[wikipedia:Parasocial_interaction|Parasocial interaction]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://olemiss.edu/news/2024/07/romantic-movies-perception/index.html Romance media could alter perceptions of relationships] (University of Mississippi) * [https://www.psychologytoday.com/au/blog/finding-love-the-scientific-take/202411/rom-coms-can-a-movie-genre-impact-our-perceptions-of Rom-coms : Can a movie genre impact our perceptions of love?] (Psychology Today) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] po28uws8diiny7q7mj88ldidu5dy12c 2835648 2835634 2026-10-03T08:26:54Z U3247927 3005952 included a table 2835648 wikitext text/x-wiki {{title|Romantic entertainment and love beliefs: How does romantic entertainment influence beliefs and expectations about love and romantic relationships?}} __TOC__ == Overview == {{RoundBoxTop|theme=3}} [[File:Couple cuddling on the couch with a movie.png|right|250x250px]] '''Scenario''' After a{{g}} long day, Adam and Eve settle onto the couch to watch {{g}} a romantic comedy called [[wikipedia:Pretty_Woman|Pretty Woman]]. Throughout the film, they are captivated by the unlikely romance, admire the grand gestures of affection and enjoy the story's happy ending. As the credits roll, Eve jokingly says that "Romantic movies make relationships looks so effortless. I wish real life worked like that" {{RoundBoxBottom}} Although the comment is playful, it raises an important question: How does romantic entertainment influence beliefs and expectations about love and romantic relationships ? As romantic films, television programs, music and media have become increasingly accessible, researchers have questioned whether repeated exposure shapes people's beliefs about how love should develop and how relationships should function. Since the 1930s, romantic films have consistently been among the most popular genres. Between 1995 and 2010, romantic comedy genre was the sixth highest grossing category of films, generating over $10 billion. The popularity of these films has led some academics to theorise on why such films had a large appeal. Academic Galician (2004) argued that people seek romantic content in the media in order to see relationships work despite all obstacles. Individuals seek out content learn about dating and romance (Hefner & Wilson, 2013). Reality TV has also flourished in recent decades and is produced to suit almost all demographics. Many people tune in because they perceive contestants and their reactions to be authentic or even relatable. Reality dating and dating competition shows such as The Bachelor and Love Island, isolate their participants and encourage the formation of romantic and platonic relationships in absence of familiar people or surroundings (Beaty, 2021). Romantic entertainment is a common source of information about love and relationships, yet the messages it promotes may not always reflect the reality of a relationship. Understanding how these portrayals influence beliefs and expectations can help individuals engage with romantic media more critically. This chapter explores how romantic entertainment influences beliefs and expectations about love and romantic relationships. {{RoundBoxTop|theme=3}} '''Focus questions''' * What are romantic beliefs and relationship expectations and why are they important? * What romantic ideals and relationship messages are commonly portrayed in romantic entertainment? * How do psychological processes such as romantic script, parasocial attachment and social comparison shape audience beliefs / perception about love and relationships? * What are the potential benefits and drawbacks of using romantic entertainment as a source of relation ship knowledge? {{RoundBoxBottom}} == Romantic beliefs and relationship expectations == ==== What is love? ==== [[wikipedia:Love|Love]] is a fundamental aspect of the human experience, one of the most crucial components of intimate relationships. According to Sternberg's triangular theory of love, romantic love consists of three core components: intimacy, desire and decision/ commitment, the intensity of the points varies as a function of relationship duration (''Sorokowski'' ''et al., 2021)''. Romantic love is a nearly universal phenomenon that contributes to human evolution and happiness. Love in romantic relationships is commonly defined as passionate and erotic love, distinguished from love in other types of relationships (parental and friendship) ''(Chen, Xia & Dunne, 2024)'''.''''' ==== Romantic relationships ==== Romantic relationships are a key contributor to overall wellbeing. Forming and maintaining intimate partnerships is an important component of our health and happiness. However there are many factors that affect the likelihood and maintaining a healthy relationship, personality, mental health, beliefs held about relationships. ==== Romantic ideals / romanticism ==== Romantic ideals refer to a broad set of beliefs about the power of love and what constitutes a successful romantic relationship (''Bell, 1975, as cited in Sprecher & Metts, 1989; Knox & Sporakowski, 1968, as cited in Sprecher & Metts, 1989''). The romantic ideal, originally proposed by Lantz (1968), reflects culturally endorsed beliefs within Western societies about how romantic relationships should form, develop, function, and be maintained. These beliefs represent shared cultural understandings of romance rather than expectations directed towards a specific partner. According to Sprecher and Metts (''1989''), romantic love is characterised by beliefs such as love at first sight, the existence of a single true love, the notion that love can overcome all obstacles, the idealisation of romantic partners, and the belief that individuals should follow their hearts when choosing a partner. The psychological mechanisms underlying this influence are discussed later in the chapter. More broadly, research commonly identifies four central themes associated with romantic ideals: the belief that love can overcome any obstacle ("love finds a way"), the existence of a soulmate ("one and only"), the idealisation of a romantic partner, and the belief in love at first sight (''Hefner & Wilson, 2013''). * '''Love finds a way''' * '''One and only''' * '''Idealisation''' * '''Love at first sight''' Romantic ideals reflect endorsement of a broader romantic ideology rather than feelings towards a specific individual. These beliefs represent shared assumptions about how love and relationships should develop and function, which individuals may hold before entering a romantic relationship (Sprecher & Metts, 1989). == Romantic ideals portrayed in entertainment == Romantic ideals are shaped by personal experiences and broader cultural influences; however these beliefs do not develop in isolation. Entertainment play a significant role in shaping and reinforcing romantic ideals by exposing audiences to idealised portrayals of love, intimacy, and relationship development. Across a range of media formats, romantic entertainment commonly presents idealised depictions of love and relationship development, promoting messages that may differ from the realities of everyday romantic relationships. {| class="wikitable" |+Table 1. Common romantic ideals portrayed in romantic entertainment media !Romantic ideal !Description |- |Soul mate / one and only |Belief that there is one perfect partner destined for each person |- |Love at first sight |Belief that genuine love can occur immediately upon meeting |- |Idealisation of partner |Expectation that romantic partners are nearly flawless |- |Love conquers all |Belief that love can overcome obstacles and challenges |} The influence of romantic entertainment extends beyond romantic comedies. Disney Princess films frequently incorporate romantic narratives that reinforce many of the same beliefs found in romantic comedy films, including soulmate ideals, idealisation of partners, and the expectation that true love leads to a happy ending (Hefner & Kretz, 2021). Beyond serving as a source of relationship knowledge, romantic entertainment may also attract audiences through emotional investment in fictional relationships. van Monsjou and Mar (2019) suggest that engagement with fictional romances provides opportunities for individuals to explore romantic experiences, develop understandings of intimacy, and reflect on relationship possibilities without directly experiencing them. Consequently, fictional relationships may serve both entertainment and psychological functions by allowing audiences to engage with romantic themes and relationship dynamics. Romantic films / rom coms * soulmates * grand gestures * destiny * love conquers all * attractive partners * conflict resolved quickly * overcoming obstacles * happily-ever-after ending === Reality dating television === * accelerated relationship development * attraction and compatibility * dating as entertainment === Popular music === * love songs reinforce cultural ideas about passion, intimacy and commitment These findings suggest that media consumption may influence how individuals interpret and apply romantic messages to their own relationships. == Psychological mechanisms of media influence == === Cultivation theory === === Social cognitive theory === Suggests that human behaviour is determined by environmental influences and internal dispositions. Individuals are more likely to imitate models who are similar to themselves.<quiz display="bold"> {A woman regularly watches romantic television shows in which husbands are consistently affectionate, attentive and express love through grand gestures. Over time, she becomes dissatisfied with her own husband's lower level of affection. According to Social Exchange Theory, what best explains her dissatisfaction?: |type="()"} - She has developed a parasocial relationship with the fictional husband + She is comparing her relationship to the idealised rewards and expectations portrayed in the media - She is experiencing attachment insecurity caused by watching romantic television </quiz> === Romantic scripts and schemas === '''Social exchange theory (SET)''' Social exchange theory provides insight into how media-shaped expectations may affect relationship satisfaction. The theory proposes that individuals evaluate romantic relationships by comparing perceived rewards, such as affection, companionship, and emotional support, against perceived costs, including conflict, compromise, and lost alternatives. Romantic entertainment often emphasises the rewards of relationships while minimising their costs, potentially encouraging unrealistic expectations. As a result, individuals may evaluate their own relationships against idealised media standards and experience dissatisfaction when reality fails to meet those expectations (Banjo, 2002. == Benefits and drawbacks of romantic entertainment == Although it may seem intuitive that holding such romanticised and idealistic beliefs would lead to worse relational outcomes, research actually contradicts this conclusion. The evidence suggests that embracing romanticism can have both positive and negative consequences. A positive effect is that these beliefs may facilitate to the development of romantic relationships because romanticism can act as a rose-coloured filter through which people view early relationship experiences (Lippmann et al., 2014). === Potential benefits === === Potential drawbacks === ==== Social Exchange Theory and relationship expectations ==== Use this heading structure: * [[#Overview|Overview]] * * What is love * * * 3 to 6 major headings tailored to the topic; can have sub-headings, but: ** avoid having only one sub-heading ** provide an introductory paragraph before breaking into sub-sections * [[#Conclusion|Conclusion]] * See also * References * External links Even Disney princess animated genre could be an important source of influence on young viewers perception of reality more specifically their beliefs regarding romantic relationships. Nearly all Disney princess movies feature a romantic component (Hefner & Kretz, 2021). Following the same beliefs as romantic comedy films. ==Key points== For the topic development, for each heading and sub-heading: * Provide at least three bullet-points, including for the Overview and Conclusion * Include key citations ==Figures== [[File:ChatGPT Image of couple watching romantic media together.png|thumb|140x140px|'''Figure 1.''' A couple watching a romantic comedy, illustrating exposure to romantic entertainment. ]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etc. * Embed figures throughout the chapter, starting with the scenario in the Overview section * Caption figures (use '''Figure #'''. and explain the relevance of the image to the text) * Images must be embedded from [[commons:|Wikimedia Commons]] * Images can be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Cite each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios, case studies, or examples describe concepts in action * Can be real or fictional; if real, provide citations * Can be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * Present using [[#Feature boxes|feature boxes]] {{anchor|Feature box}} ;Feature boxes * Highlight key content using [[Motivation and emotion/Wikiversity/Feature box|feature boxes]], but don't overuse, otherwise they lose their effect * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Embedded links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") {{anchor|Tables}} ;Tables * Use to tables to organise and summarise information * Cite each table at least once in the main text (e.g., see Table 1) * Tables should be captioned * [[Motivation and emotion/Wikiversity/Tables|More example tables]] which can be adapted '''Table 1''' A Descriptive Caption Which Explains The Table Contents and its Relevance to the Text e.g., The 2 x 2 Johari Window Model Showing Knowing x Self/Other {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two quiz questions for each main section is better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia. Ask about important information such as take-home messages * Ask easy rather than hard questions * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {The purpose of quizzes is to provide an interactive learning feature: |type="()"} + True - False {Long and complex quiz questions are recommended: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Draft clear take-home message(s), even at the topic development stage * Together, the [[#Overview|Overview]] and Conclusion should summarise the problem, its significance, and how psychological science contributes to understanding and addressing it * Recommended length: 150 to 330 words === To what extent should people use romantic entertainment as a guide for real-life relationships? === This is essentially your concluding discussion{{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [[wikipedia:Soulmate|Soulmate]] (Wikipedia) * [[Motivation and emotion/Book/2011/Love|Love]] (Book chapter, 2011) * [[wikipedia:Parasocial_interaction|Parasocial interaction]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Banjo, O. O. (2002). The effects of media consumption on the perception of romantic relationships. Penn State McNair Journal, 9(1), 9-33. Beaty, L. S. (2021). It’s Love Island, Not Friend Island”: Authenticity and Surveillance in Reality TV, a Literature Review and Content Analysis “. WWU Honors College Senior Projects 456. Chen, Y., Xia, M., & Dunne, S. (2024). Romantic Love is Not Only “Romantic”: A Grounded Theory Study on Love in Romantic Relationships. The Journal of Psychology, 158(1), 64–83. https://doi.org/10.1080/00223980.2024.2305442 Hefner, V., & Kretz, V. E. (2021). Does the Glass Slipper Fit?: Disney Princess Films and Relationship Beliefs and Attitudes. Journal of Media Psychology, 33(3), 125–133. https://doi.org/10.1027/1864-1105/a000290 Hefner, V., & Wilson, B. J. (2013). From Love at First Sight to Soul Mate: The Influence of Romantic Ideals in Popular Films on Young People’s Beliefs about Relationships. Communication Monographs, 80(2), 150–175. https://doi-org.ezproxy.canberra.edu.au/10.1080/03637751.2013.776697 Lippman, J. R., Ward, L. M., & Seabrook, R. C. (2014). Isn’t It Romantic? Differential Associations Between Romantic Screen Media Genres and Romantic Beliefs. Psychology of Popular Media Culture, 3(3), 128–140. https://doi.org/10.1037/ppm0000034 Knee, C. R. (1998). Implicit Theories of Relationships: Assessment and Prediction of Romantic Relationship Initiation, Coping, and Longevity. Journal of Personality and Social Psychology, 74(2), 360–370. https://doi.org/10.1037/0022-3514.74.2.360 Sorokowski, P., Sorokowska, A., Groyecka, A., Aavik, T., Akello, G., Alm,C., Amjad, N., Anjum, A., Asao, K., Atama, C. S., Atamtürk Duyar, D., Ayebare, R., Batres, C., Bendixen, M., Bensafia, A., Bizumic, B., Boussena, M., Buss, D. M., Butovskaya, M., … Sternberg, R. J. (2021). Universality of the Triangular Theory of Love: Adaptation and Psychometric Properties of the Triangular Love Scale in 25 Countries. The Journal of Sex Research, 58(1), 106–115. https://doi.org/10.1080/00224499.2020.1787318 Sprecher, S., & Metts, S. (1999). Romantic Beliefs: Their Influence on Relationships and Patterns of Change Over Time. Journal of Social and Personal Relationships, 16(6), 834-851. https://doi.org/10.1177/0265407599166009 Van Monsjou, E., & Mar, R. A. (2019). Interest and Investment in Fictional Romances. Psychology of Aesthetics, Creativity, and the Arts, 13(4), 431–449. https://doi.org/10.1037/aca0000191 }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://olemiss.edu/news/2024/07/romantic-movies-perception/index.html Romance media could alter perceptions of relationships] (University of Mississippi) * [https://www.psychologytoday.com/au/blog/finding-love-the-scientific-take/202411/rom-coms-can-a-movie-genre-impact-our-perceptions-of Rom-coms : Can a movie genre impact our perceptions of love?] (Psychology Today) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Love]] [[Category:Motivation and emotion/Book/Media]] [[Category:Motivation and emotion/Book/Relationships]] kvl9bxrpu9h5jg15f3o7tqmoqtwr2ez 0 331105 2835603 2835283 2026-10-03T03:05:38Z U3261376 3106964 first category 2835603 wikitext text/x-wiki {{title|Sex work motivation:<br>What motivates sex work and how does this impact worker experiences?}} __TOC__ == Overview == {{RoundBoxTop|theme=5}} [[File:“Sex workers rights are human rights” - Europride 2019.jpg|thumb|200px|'''Figure 1'''. Prostitution in Sao Paulo, Brazil]] '''What would you do if you found yourself in this scenario?''' Feeling lost and left with little money of your own, but you have to make ends meet. What will you do when you have to pay your rent? What if you have a child? Would you turn to sex work to make ends meet? If you did... would that make your rights to a safe and trustworthy work environment just disappear? Sex work itself is illegal. Having been driven to such means to make ends meet, you must register as self-employed to pay taxes, buy a house, or even just be considered an employee. [https://doi.org/10.1080/14427591.2020.1830841. (Huglstad et al., 2020)]. Sex workers rights activists proclaim that sex work is work, that they are not victims in need of help, and that they are entitled to any basic human right (see Figure 1). That in itself poses another scenario: What drives people into sex work? Is it finance and convenience? Or is it pleasure and fun? {{RoundBoxBottom}} [[Sex education|Sex]] work or [https://www.jstor.org/stable/pdf/10.1086/324390.pdf?casa_token=lfbXJZ0pqcQAAAAA:Y2KROCBvBcyTR_OkV9GU9nGAnh5lp0VKsrro0wiLqGF_MeXybi6-z3pvrxj97OM6GMvGv6r_Zbmt_s9RvHKDiUw3JzDhAPWvsSz3uzNI4nXa_mNdM5o prostitution] is a broad topic which includes many elements of sexual activities. In 2005, Harcourt defined "prostitution, [as] the provision of sexual services for money or its equivalent," ([https://pmc.ncbi.nlm.nih.gov/articles/PMC1744977/pdf/v081p00201.pdf Harcourt, 2005]). To understand sex work as a concept it is important to first understand why people may turn to sex work for an income to begin with. This topic has been discussed at nauseum however often in a negative and clouded light. As stated in The Journal of Sex Research in 2013, it wasn't until the 1990's that literature that looked into this topic stopped looking at prostitutes as "predominantly... ‘‘evil characters’’ or ‘‘sick personalities.’’" ([https://www.tandfonline.com/doi/pdf/10.1080/00224499.2012.696285?casa_token=wgasXeCYwFAAAAAA:0Iuwx4QzcbYP0ZAj6GISEX5oDHr4t-phuo4VUxmwfMIVjTrLXaTBhIoYpXtC9r0WaeYaxII3ObSZ Vanwesenbeeck, 2013]) and carrying a lense of ""simmering abhorrence, incomprehension, and fascination" (Vanwesenbeeck, 2013) towards these individuals. Some of the main concepts that will be unpacked below are what motivates people to go into sex work, answers are commonly, "financial, sexual, recreational motives, and coercion (i.e., [https://www.jstor.org/stable/pdf/30048506.pdf?casa_token=pqlPp6XrnF0AAAAA:SFSvr6r_WKbLtO638dPPYk35oZFz9anPcub1Be7iUjk9AMX7a4h73yPLMJ6efSSxWVK7Az4Upc3UgohFYtht3QViGNIyTXPWCkBaY5lhRS_RGNFzUoA trafficking])" (Vanwesenbeeck, 2013), these experiences can often lead to negative workplace satisfaction. External influences like the adult entertainment industry and whorephobia can also heavily influence an individual's workplace satisfaction when working in the prostitution industry. To learn the in depth relationship between motivation to join sex work and the quality of said workplace satisfaction continue reading below! {{RoundBoxTop|theme=5}} '''Focus questions''' * What are the main driving forces causing people to turn to sex work? * What is the effect of the adult entertainment industry on sex workers? * What is the condition of work place satisfaction for sex workers? * How does whorephobia and others' actions towards sex workers effect their work place? * Do those who turn to sex work for pleasure have different experiences to those who do so for money alone? {{RoundBoxBottom}} == Driving forces pushing people towards sex work == There are many driving forces that can influence someone's motivation to enter any kind of work force. Unfortunatly financial motives are often what pushes people to turn to the sex trade. Although there is evidence that some do find pleasure and positive experiences through sex work. So, those who turn to sex work for pleasure have different experiences to those who do so for money alone? === Intrinsic motivation === It would be irresponsible to look at motivation without first adressing key theories that may explain why a person would perform any behaviour. By looking at [[Intrinsic motivation|intrinsic motivational]] forces that may influence someone into the sex trade it can give a fuller explination of their actions. What internal need are they fulfilling by performing these actions? Intrinsic motivation falls under the "definition of [an] action as rewarding in itself" (Fishbach & Woolley, 2021), not nesseserily the outcome/ final goal. Although economic needs are a common factor of joining the sex trade — and is considered an extrinsically motivating factor — there is "research confirming that individuals sometimes enter this occupation for reasons of personal fulfillment"(Arnold et al., 2020) which more closly alligns into intrinsic motives. === Extrinsic motivation === Influences that effect people on an external level, or [[Extrinsic motivation|extrinsic motivations]], are influences that effect people outwardly, forces that is harder for an individual to control. A common driving factor for joining sex work, as stated above, is earning money. === Maslow's hierarchy of needs === [[File:Maslow's hierarchy of needs.svg|thumb|290x290px|'''Figure 1'''. Maslow's Heirarchy of Needs]] - looking at maslows hierarchy of needs, what may cause people to turn to sex work? - does peoples standing on the hierarchy influence their workplace satisfaction? == How motivations to join the sex work industry differ from men to women == {| class="wikitable" |+Percentages showing [https://www.annualreviews.org/docserver/fulltext/psych/65/1/annurev-psych-010213-115057.pdf?expires=1790841225&id=id&accname=au&checksum=112378A7023548A83B7DDDA7288D0EA0 gender differences] in paying for vs working in sex ! !Participate in sex work !Pay for sex work |- |Men |20% |12% |- |Women |80% |1% |} === Male sex workers === - why are they less prevelent in the public eye? - why are they not degraded as much as female workers? - do men turn to sex work to fulfill their own physical needs? === Female sex workers === - why are they more prevelent in the public eye? - why are they degraded more then male workers? - do women turn to sex work to fulfill their own physical needs?<quiz display="simple"> {Knowing what motivates people to go into sex work, if you found yourself in their shoes would you do it too?: |type="()"} + Yes - No </quiz> == The adult entertainment industry and its relationship to sex workers == - does this carer add unrealistic expectations towards sex workers and what people will ask to recieve? - looking into detail about how the adult entertainment industry can glorify sex work and how that may motivate people to join the industry. == Work place satisfaction for sex workers == - is there an element of sexual satisfaction which motivates sex workers? - is there a difference between men and women? - how to sexworkers respond to violence in their line of work and how may that motivate their actions? == Whorephobia and how it influences work place satisfaction == - how do unfavorable opinions of sex workers effect motivation and how does that relate to workplace satisfaction? "on a continuum from amotivation through extrinsic to intrinsic motivation. We consider this as an individual difference variable that changes the relationship between perceived physical danger and desire to leave the sex work occupation." (Arnold et al., 2020) - how does the prevelence of this violence motivate sex workers to continue in their chosen carer path? - how does [https://www.jrc-concordia.ca/wp-content/uploads/2017/01/Journal-of-Religion-and-Culture-Vol-27-no-1.pdf#page=13 whorephobia] effect workplace satisfaction?<quiz display="simple"> {More men then women experience sexual violence as a sex worker: |type="()"} - True + False {There are more women in the sex trade then there are men: |type="()"} - True + False </quiz> ==Conclusion== Through this chapter we have spoken about the different motivations that drives people into sex work and how that can influence work place experiences. We have done this by assesing how age and gender can influence motivations as well as what circumstances can effect those drives. - summerise the different motivations that lead to sex work, considering gender and age. - summerise how socioeconomic factors influence sex worker experiences, including prejudices. * Recommended length: 150 to 330 words {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[Extrinsic motivation|Extrincis motivation]] (Wikiversity) * [[Intrinsic motivation]] (Wikiversity) * [[Maslow's hierarchy of needs]] (Wikiversity) * [[Sex education]] (Wikiversity) * [[Motivation and emotion/Book/2015/Sex worker motivation|Sex work motivation]] (Book chapter, 2015) ==References == {{Hanging indent|1=Arnold, K. A., Turner, N., Barling, J., & Iverson, R. D. (2020). Feeling safe while doing sex work: Motivation for entering sex work moderates the relationship between perceptions of physical danger and desire to leave sex work. Safety science, 129, 104795.[https://pdf.sciencedirectassets.com/271730/1-s2.0-S0925753520X00063/1-s2.0-S0925753520301922/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEDgaCXVzLWVhc3QtMSJIMEYCIQDLwNUlkBLTVH2W9sUG%2FL8xBiniafqj2YjDThRYfM8j7gIhAPpb6usou%2FCfun2GlN94HXvsSkZFNyb27ndcS5LYkmOqKrIFCAEQBRoMMDU5MDAzNTQ2ODY1IgwVqu3mdLMxIbQQT4EqjwVyEzAMhHQLz8i1mRyce46T7kq8nFGMPfsWbvGbIjwD5sTztgejH2H6uoazHE3ceLn%2BHJAMU2gcjulIYqlqCSyvPuWe%2BScfjOHfBfQZnctFdUl2oWqaahIi9YzAVcTWeftoZk%2FylcCfLerGklhqD7HwVv2vZdBzsx94xmqQ1V1uUbIbAIkJ30NfSj00z1cSaUL1q96wM120kL%2By3Fd%2FYRnMyLzgn343yVJM%2BYWksKgOiezOVEa%2BhmIbNUYzcpb%2Fkcah8UWIgZWXjdLp2fZTJB2yCYbj091PqbNLxbKvPvqaCnGI%2BSTaCPpupfKIdR02m2jepxiZI8%2FZrLGy71f0vYzU%2Be120AQZZCPdjTZZAOP9xB%2Bvfpp48EyclSYmN0rSn2bZcMSiqosJl%2Bz6BiiW3J9SmbZXGNWeECGltEiG%2FP%2FRiZ0uB94c2nIGVzBsfJ4fe7UAD3mlWR9UBCT5FirlPE3Dzap%2BH1xnOe%2BGN481xmH1YSoL3FACDW8J5HUez%2BURv5vTzFDq4nON4cFWhMdgPNfu8v%2BjU35utqWccWAfsows5CFE%2FdiNDUQa7EI1XwF00wtRp2k06DjOWBl%2FggL2duXPWFHHaJnOEovHXAGzQUl%2FzBocvEbkU%2BstJU3AjUwCqbrDZRVZay0dd7OLQ6NCZTFxvZrhEoOwYukzgYt02XtXs23oNKc8QK8jPBsjSx8K1Xj9V9tPB5eBTptias5DuJnme5GPMo0H0rehhAFIbH1iz5aLEJ2WFAs2YMajNB9b8K9hsdcsufqw3IWvDgGjJl491SPFYv4AkHpfF%2B2OEEmH9hDMp3rnzv686gQ5u2%2FmSKrDsChJ2JSpJUuTnHdHBZrk%2Fwzx%2BpsfpZ4zLVgKx0K6MJOGtdQGOrAB7mHiroFpJKweD4wGdVv4twWKj65wn3ZS715WCDMXnol3ZjKerxKwtTiAS8MYNXREK5yRc6R%2BCtHqgkfO1QIL6y35fyD0iOWFO6ZYrI1IePcEupQe4Ao%2B8i5%2Bz9PydRXrVVCjz1A9o6ZcLX9dN59jVDboJOpZsOpXxuErU9LEKNKGTqVoHfJq%2FpKSdZkltmLsDvSAgBZORP2gpikUUQyl6A0CxzJ3SGD1ZrqXN39XlWY%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20260825T084050Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYZDNOMNKH%2F20260825%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=222054776072dd684a9220ca5e137f808570de2f4b46a74fc1fac36b8741d31e&hash=9077aea1050680fe483f80515f312051569ddf0b7a50ba90c1511bc6ef136d06&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0925753520301922&tid=spdf-2823bc10-4a39-4f2b-8a32-7409653a6f56&sid=6bbf6c2f3a5ff545d91b09f2929365875304gxrqa&type=client&tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&rh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&ua=1b130057565d500d535050&rr=a3094b5109a37504&cc=au] Fishbach, A., & Woolley, K. (2021). Annual Review of Organizational Psychology and Organizational Behavior The Structure of Intrinsic Motivation. https://doi.org/10.1146/annurev-orgpsych-012420- ‌ Footer, K. H. A., White, R. H., Park, J. N., Decker, M. R., Lutnick, A., & Sherman, S. G. (2020). Entry to Sex Trade and Long-Term Vulnerabilities of Female Sex Workers Who Enter the Sex Trade Before the Age of Eighteen. Journal of Urban Health, 97(3), 406–417. https://pmc.ncbi.nlm.nih.gov/articles/PMC7305278/ Harcourt, C. (2005). The many faces of sex work. Sexually Transmitted Infections, 81(3), 201–206. https://doi.org/10.1136/sti.2004.012468 ‌ Huglstad, M., Halvorsen, I. L. I., Jonsson, H., & Nielsen, K. T. (2020). “Some of us actually choose to do this”: The meanings of sex work from the perspective of female sex workers in Denmark. Journal of Occupational Science, 29(1), 1–14. https://doi.org/10.1080/14427591.2020.1830841. Ine Vanwesenbeeck (2013) Prostitution Push and Pull: Male and Female Perspectives, Journal of Sex Research, 50:1, 11-16, https://doi.org/10.1080/00224499.2012.696285 Maslow, A., & Lewis, K. J. (1987). Maslow's hierarchy of needs. Salenger Incorporated, 14(17), 987-990. https://www.justiceacademy.am/assets/uploads/Maslow.pdf McNamara, R. (2016). The trouble with whorephobia. Journal of Religion and Culture, 27(1), 13-34. McKeever, N. (2025). Is Sex Work Inherently Gendered? Hypatia, 40(3), 633–652. doi:10.1017/hyp.2024.56 ‌Redwood, R. (2018). Myths and Realities of Male Sex Work. Red light labour: Sex work regulation, agency, and resistance, 167-180.[https://books.google.com.au/books?hl=en&lr=&id=8GJoDwAAQBAJ&oi=fnd&pg=PA167&dq=Myths+and+Realities+of+Male+Sex+Work&ots=mQOUglmuLU&sig=XHi0ybe124LABeHMulfVUMeYsxw#v=onepage&q=Myths%20and%20Realities%20of%20Male%20Sex%20Work&f=false] Sex worker - Wikipedia. (2023). Wikipedia.Org. https://en.wikipedia.org/wiki/Sex_worker}} ==External links== [https://www.jstor.org/stable/pdf/10.1086/324390.pdf?casa_token=lfbXJZ0pqcQAAAAA:Y2KROCBvBcyTR_OkV9GU9nGAnh5lp0VKsrro0wiLqGF_MeXybi6-z3pvrxj97OM6GMvGv6r_Zbmt_s9RvHKDiUw3JzDhAPWvsSz3uzNI4nXa_mNdM5o A Theory of Prostitution] (The University of Chicago Press) [https://www.annualreviews.org/docserver/fulltext/psych/65/1/annurev-psych-010213-115057.pdf?expires=1790841225&id=id&accname=au&checksum=112378A7023548A83B7DDDA7288D0EA0 Gender Similarities and Differences] (The Annual Review of Psychology) [https://www.jstor.org/stable/pdf/30048506.pdf?refreqid=fastly-default%3Abdd06e603789b9df6429bacfd93907fe&ab_segments=&initiator=&acceptTC=1 Human Trafficking] (JSTOR)‌ [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Sex]] [[Category:Motivation and emotion/Book/Work motivation]] d9edue4n5lgyyaa3zotcd5h51zhbzne 0 331144 2835601 2835459 2026-10-03T02:06:50Z U3246286 3105520 focus question one 2835601 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024). Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness, and relatedness by feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs, autonomy, competence, and relatedness to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments could either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, whilst also having the opportunity of caring for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported which promotes overall psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked which can contribute to distress, maladaptive behaviours and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous of controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). === The Military Environment === The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == * Psychology need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. * SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. === Autonomy and the experience of choice === * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] 2covp695rhw3fiuhb587z8c0sdiplwb 2835604 2835601 2026-10-03T03:08:38Z U3246286 3105520 finished focus question 1 2835604 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024). Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness, and relatedness by feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs, autonomy, competence, and relatedness to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments could either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, whilst also having the opportunity of caring for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported which promotes overall psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked which can contribute to distress, maladaptive behaviours and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous of controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == * Psychology need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. * SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. === Autonomy and the experience of choice === * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] ql6ec4phequ2mxtf7sgabtru6ahtnn0 2835605 2835604 2026-10-03T03:38:27Z U3246286 3105520 2835605 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity, and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. '''These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).''' Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life, and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness refers to feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment, and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development, or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs; autonomy, competence, and relatedness, to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling influenced, coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes, and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, as well as having the opportunity to care for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported, which promotes overall positive psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked, which can contribute to distress, maladaptive behaviours, and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous to controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment that occurs during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs, it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == * Psychology need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. * SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. === Autonomy and the experience of choice === * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] q2w2lkmjbx07o71utwod1efgzsu5x3u 2835621 2835605 2026-10-03T05:10:55Z U3246286 3105520 2835621 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity, and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. '''These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).''' Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life, and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness refers to feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment, and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development, or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs; autonomy, competence, and relatedness, to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling influenced, coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes, and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, as well as having the opportunity to care for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported, which promotes overall positive psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked, which can contribute to distress, maladaptive behaviours, and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous to controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment that occurs during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs, it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == Psychological need satisfaction provides a mechanism through social environments which can influence wellbeing, motivation, and psychological adjustment. need satisfaction provides a mechanism with social environments may either positively or negatively alter their wellbeing, motivation, and psychological adjustment during their transition. * SDT conceptualises motivation along a continuum of self-determination, it ranges from intrinsic motivation and forms of autonomous extrinsic motivation to then controlled forms of extrinsic motivation and amotivation. === Autonomy and the experience of choice === * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] dc770tjjj7lgk2daxhfbefls4dt2zb4 2835635 2835621 2026-10-03T06:17:29Z U3246286 3105520 focus question 2 2835635 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity, and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. '''These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).''' Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life, and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness refers to feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment, and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development, or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs; autonomy, competence, and relatedness, to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling influenced, coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes, and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, as well as having the opportunity to care for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported, which promotes overall positive psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked, which can contribute to distress, maladaptive behaviours, and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous to controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment that occurs during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs, it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == Psychological need satisfaction provides a mechanism through social environments which can influence wellbeing, motivation, and psychological adjustment. SDT's value for understanding military-to-civilian transition is in it's ability to explain why similar transitions might produce different psychological experiences. Instead of assuming that leaving the military is inherently positive or negative, SDT proposes that adjustment mainly depends on whether veterans experience their new environments as either supportive or frustrating of their psychological needs. === Autonomy and the experience of choice === Autonomy refers to an individual's experience of self-directed and volitional action. They perceive their behaviour as personally aligned with their personal interests, values, and sense of self. It's important to note that structured environments do not necessarily undermine autonomy, but instead can be satisfied when individuals understand the purpose of an activity and endorse their own actions. * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] owzkis0we1b96aitbc6akiqoeu3if35 2835638 2835635 2026-10-03T06:36:49Z U3246286 3105520 2835638 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity, and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. '''These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).''' Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life, and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness refers to feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment, and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development, or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs; autonomy, competence, and relatedness, to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling influenced, coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes, and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, as well as having the opportunity to care for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported, which promotes overall positive psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked, which can contribute to distress, maladaptive behaviours, and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous to controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment that occurs during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs, it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == Psychological need satisfaction provides a mechanism through social environments which can influence wellbeing, motivation, and psychological adjustment. SDT's value for understanding military-to-civilian transition is in it's ability to explain why similar transitions might produce different psychological experiences. Instead of assuming that leaving the military is inherently positive or negative, SDT proposes that adjustment mainly depends on whether veterans experience their new environments as either supportive or frustrating of their psychological needs. === Autonomy and the experience of choice === Autonomy is not always reduced by structure or externally defined expectations, instead, it can actually be supported when individuals understand and agree with the purpose behind an activity and personally endorse their own actions. This is particularly relevant to military context as structured expectations and roles can coexist with a sense of personal choice when individuals understand and accept the reasons for their actions. Autonomy refers to an individual's experience of self-directed and volitional action. They perceive their behaviour as personally aligned with their personal interests, values, and sense of self. It's important to note that structured environments do not necessarily undermine autonomy, but instead can be satisfied when individuals understand the purpose of an activity and endorse their own actions. * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] 68t0q1rbz4fi8xa9g1u7lew0iuqgqrs 2835651 2835638 2026-10-03T08:31:21Z U3246286 3105520 2835651 wikitext text/x-wiki {{title|Self-determination theory and military veteran reintegration:<br>How do autonomy, competence, and relatedness shape psychological adjustment after military service?}} __TOC__ ==Overview== {{RoundBoxTop|theme=5}} ;Scenario [[File:Military-to-civilian reintegration.png|thumb|'''Figure 1'''. Transition from military service to civilian life]] What happens when the thing that gave your life structure suddenly vanishes? For many military veterans, leaving the service gains them freedom, but freedom can also bring uncertainty. After 15 years of military service including posting and deployments, a veteran returns home expecting civilian life to feel like freedom (see Figure 1) . Instead, freedom feels disorientating. For over a decade, their days were shaped by established routines, explicit expectations, clear responsibilities, a sense of belonging and being part of a team. Roles were understood. Decisions had purposes. Belonging was built into day-to-day life. The team was your support. Then, almost, overnight, the rules had changed. There are no longer orders telling them what needs to be done, no team waiting or relying on them, and no familiar structure defining what it means to be successful. Decisions that were once straightforward, now require navigating through uncertainty and often with little guidance or support. The confidence they had developed through military service may not cleanly translate into civilian life, whilst the relationships that brought connection and belonging may suddenly be hundreds or thousands of kilometres away. Disconnection is real. What was supposed to be liberation has now become a profound psychological adjustment. Self-determination theory provides an understanding of why. If military life has shaped how a veteran experiences autonomy, relatedness and competence {{ic|APA style uses serial commas}} , what happens when the life that once supported or constrained these needs is removed? {{RoundBoxBottom}}Military-to-civilian [wikilink military and civilian] transition is a major period of adjustment that brings an increased likelihood of psychosocial risk. This transition involves simultaneous changes to daily structure, social relationships, occupational roles, their identity, and sources of purpose. These changes can challenge veteran's wellbeing, motivation, and psychological adjustment as they adapt to a new environment. '''These transitions can have long-term implications to their wellbeing with 78% of Australian defence members experienced a difficult transition and 50% were still struggling after 10 years (Romanik, et al. 2024).''' Self-determination theory (SDT) provides a framework for understanding psychological adjustment by suggesting that autonomy, competence, and relatedness are basic psychological needs that are fundamental for wellbeing, quality of life, and psychological flourishing. Autonomy involves experiencing a sense of volition and choice, competence refers to feeling satisfactory and capable in their skills and relatedness refers to feeling connected to and valued by others (Ryan & Deci, 2022). Satisfaction of these needs support more autonomous motivation and positive psychological outcomes, whereas, frustration can undermine both wellbeing and motivation. This chapter applies SDT to military-to-civilian reintegration, it explores how military experiences and their transition to civilian life can either support or frustrate veterans' autonomy, competence, and relatedness. It'll examine how changes in these psychological needs may shape or influence veterans' emotional wellbeing, psychological adjustment, and motivation. Finally this chapter will consider how SDT-informed interventions can both support need satisfaction and facilitate a successful reintegration. * '''SDT as the theoretical framework [introducing main theory that will be discussed throughout chapter]''': * SDT is a useful framework and provides a key blueprint for understanding the motivational basis of personality and social behaviour, along with the relation of basic psychological needs for wellbeing, psychological flourishing and positive quality of life (Deci, Ryan 2022). * SDT proposes that autonomy, competence, and relatedness are basic psychological needs that contribute to individuals' wellbeing and adaptive functioning. Satisfaction of these needs will contribute to positive wellbeing, whereas, frustration can contribute to negative psychological outcomes (Ryan, et al. 2022). * This chapter will therefore utilise SDT to examine how changes associated with military to civilian reintegration may influence these three psychological needs. * '''Reintegration and psychological adjustment [what do these three needs affect]:''' Following discharge (or voluntary leaving) veterans may need to reconstruct these psychological needs within unfamiliar civilian environments. * Employment difficulties may then challenge competence, whilst their loss of military relationships and identity may undermine their belonging and relatedness. Greater freedom may support their autonomy, but can also create uncertainty without their consistent military structure (Carra, et al. 2022). * This book chapter will therefore examine how these changes may shape veterans' psychological adjustment during reintegration and consider how understanding veterans' psychological needs can inform approaches/implementations to supporting successful transitions. Author note: depending on how book chapter flows along with word count, the following 'sections' may be included in the overview and structured accordingly. * '''Military service and psychological needs (Carra et al. 2022):''' transitions can have a significant adverse effect on veterans' wellbeing. The transitions usually involve simultaneous changes across multiple aspects of their life, this includes, their identity, finances, occupation, residences, routines, relationships and social support. These changes may accumulate to an increase in vulnerability to psychological distress and mental health difficulties. Research has highlighted the importance of providing suitable transition support to effectively assist veterans in navigating these changes and adapting comfortably in civilian life. {{RoundBoxTop|theme=5}} ;Focus questions {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} 1. How does Self Determination Theory {{ic|Use APA style for capitalisation of theories (lower case)}} explain psychological adjustment during military-to-civilian transition? 2. How does satisfaction or frustration of autonomy, competence, and relatedness shape veterans' motivation and emotional wellbeing? 3. How military experiences and reintegration shape autonomy, competence, and relatedness? 4. How can Self Determination Theory informed interventions support psychological need satisfaction and successful reintegration? {{RoundBoxBottom}} == How does Self Determination Theory explain psychological adjustment during military-to-civilian transition? == {{ic|The first two bullet points could make for an effective Overview}} Self-determination theory is an organismic and dialectical framework of human motivation and personality that was developed by Edward Deci and Richard Ryan. SDT offers an understanding for how motivation and psychological wellbeing are shaped by the interaction between individuals and their environments. Instead of viewing motivation as solely an individual characteristic, SDT suggests that social context can create conditions that either support healthy psychological development, or contribute to difficulties in psychological functioning. A key principle of SDT is that people require satisfaction of three basic psychological needs; autonomy, competence, and relatedness, to function effectively. === What is Self Determination Theory (SDT)? === SDT proposes that people have an inherent tendency towards development and growth, but social environments either support or interfere with this process. This theory considers both the individual and environment when explaining psychological functioning and motivation (Deci, & Ryan 2022). Autonomy is a core psychological need to experience oneself as the author of their own life, this includes having a sense of choice and self-direction in one's own actions. People can experience autonomy when their behaviour reflects their own interests, values, and choices rather than feeling influenced, coerced or controlled by external pressures. However, autonomy doesn't mean to be completely independent from others, instead, it involves experiencing one's behaviour as being personally endorsed and chosen. Competence is the need to feel both effective and capable in one's interaction with the environment. It involves feeling a sense of confidence in one's ability to manage tasks, achieve desired outcomes, and overcome different challenges. However, it's important to note that competence is not defined by an objective level of skill or an ability that someone possesses, instead, it's how capable and effective they perceive themselves as. Therefore, environments that create experiences of failure, criticism or an inability to manage demands might undermine this need. Relatedness is the feeling of being connected to and cared for by others, as well as having the opportunity to care for others. Individuals can experience feeling accepted, understood, valued, and developing a sense of belonging within a broader community. Relatedness is not just being surrounded by other people, but instead feeling a genuine connection and feeling that one has a meaningful place within a social group. When one experiences rejection, loneliness, or a loss of meaningful connections it can frustrate this need. Need satisfaction and need frustration are two ways to explain how environments can influence these psychological needs. Need satisfaction refers to when autonomy, competence and relatedness is supported, which promotes overall positive psychological wellbeing and growth. In contrast, need frustration can occur when these needs are actively undermined or blocked, which can contribute to distress, maladaptive behaviours, and ill-being. See table 1 below for a comparison of these needs. (Olafasen, et al. 2021) {| class="wikitable" |+Table 1. Need Satisfaction vs Need Frustration (Olafasen, et al. 2021) ! !Autonomy !Competence !Relatedness |- |Need satisfaction |experiencing choice, self-direction, freedom in ones' actions |experiencing mastery, capability and effectiveness in skills. |feeling a warm, secure and mutual care with others. |- |Need frustration |feeling controlled, pressured or compelled to act in an unwanted way |feeling incompetent, facing frequent failure and doubting abilities. |experiencing loneliness, alienation, and social exclusion. |} SDT also distinguishes between two forms of motivation, ranging from autonomous to controlled motivation. It also explains how these motivations relate to engagement, vitality, performance, and psychological health (Loverre, et al. 2024). Autonomous motivation includes intrinsic motivation and fully internalised extrinsic motivation, whereas, controlled motivation involves internal pressures or external demands. Therefore SDT does not only explain whether someone is motivated but also why they are motivated (Rybakovaite, et al. 2021). SDT provides a framework to understand the psychological adjustment that occurs during the military-to-civilian transition. The changes in roles, relationships, routines, and responsibilities can impact their autonomy, competence, and relatedness. Whilst a new environment might support some psychological needs, it may frustrate others. For example, increased freedom after service might provide greater autonomy, whereas, changes in roles and routines may challenge their feelings of competence. Similarly, changes in social connections may impact relatedness. Therefore, SDT suggests that psychological adjustment is influenced and effected by how well veterans' new environments support their psychological needs, rather than just focuses on the transition alone. The relationship between need satisfaction, frustration, wellbeing, and motivation is explored in the following sections. - move into focus question 3. The military environment is often characterised by routines, clearly defined roles, hierarchy, rules, expectations, collective goals and a sense of purpose, thus, creating a highly structured social environment. The military environment may support competence, autonomy and belonging through structured roles, occupation, training responsibilities, teamwork, unit cohesion and shared military identity. However, it's important to note that the hierarchy, externally imposed responsibilities and rules may create tensions with autonomy and might influence the extent to which motivation is either autonomous or controlled. *Veterans may experience feelings of cultural dislocation, emotional distress and identity disorientation which can be linked to the loss of structured military roles and routines. They may also experience social alienation when attempting to rebuild relationships, re-establish belongings, and adapt to unfamiliar social norms beyond the military sphere. The current known challenges that ADF members experience are cumulative effects of trauma and stress, moral injury to beliefs and expectations, stigma, and impaired help seeking (Kerr, et al 2022) along with the loss of relationships. * For those that may have left involuntarily often tend to experience difficulties in translating military skills into employment, and have a perceived lack of competence (Kerr, et al 2022). These challenges and difficulties may influence veterans' motivation and psychological wellbeing. * Therefore, military experiences may simultaneously constrain and support psychological needs rather than being universally positive or negative. == How does satisfaction or frustration of autonomy, competence and relatedness shape veterans' motivation and emotional wellbeing? == Psychological need satisfaction provides a mechanism through social environments which can influence wellbeing, motivation, and psychological adjustment. SDT's value for understanding military-to-civilian transition is in it's ability to explain why similar transitions might produce different psychological experiences. Instead of assuming that leaving the military is inherently positive or negative, SDT proposes that adjustment mainly depends on whether veterans experience their new environments as either supportive or frustrating of their psychological needs. === Autonomy and the experience of choice === Autonomy is not always reduced by structure or externally defined expectations, instead, it can actually be supported when individuals understand and agree with the purpose behind an activity and personally endorse their own actions. This is particularly relevant to military context as structured expectations and roles can coexist with a sense of personal choice when individuals understand and accept the reasons for their actions. Experiences that support autonomy have been associated with greater life satisfaction, wellbeing, and vitality. Throughout the military-to-civilian transition, veterans can experience a significance increase in personal choice as employment, relationships, and future goals shift towards the individuals' responsibility. However, having more choice does not automatically correlate to producing greater autonomy, as veterans may experience difficulties, uncertainty, and challenges in identifying choices that directly align with their values. ''Include study'' Therefore autonomy may support adjustment if veterans experience their post-discharge decisions as both meaningful and self-endorsed. However, if environments create pressure or undermine a sense of volition. * SDT identifies autonomy as a psychological need that concerns psychological self-direction, freedom, and experiencing one's actions as congruent and self-endorsed with their values and interests. * Autonomy refers to the feeling of psychological freedom and self-directions of one's thoughts, actions and feelings. * Military hierarchy does not inherently undermine or satisfy autonomy. Anatomy can be experienced within structured environments, by endorsing externally induced requests by clearly understanding the meaning and purpose behind the demanded activity. Autonomy satisfaction is critical human growth and has been linked to strong levels of individual wellbeing, life satisfaction and vitality (Grimell, 2024). * Autonomy frustration can occur when military personnel sense of volition is actively undermined, this could be due to micromanagement or heavy restrictions which can correlate to internal conflicts and/or a passive/resistant mindset. This contributes to a negative psychological outcomes with links to controlled motivation, mental health issues and higher discharge rates. Further, autonomy frustration can lead to reduced trust, unit cohesion and initiative (Grimell, 2024). * Whilst transition from military to civilian life, veterans may move from an environment strongly characterised by externally prescribed relationships and structured expectations/routines, therefore, they will have a greater responsibility when it comes to employment, relationships and future goals. * Notably, greater choice does not equate to greater autonomy as whilst choices may create opportunities, choice alone does not guarantee autonomy. Especially, if veterans feel pressure, lack meaningful alternatives and have a level of uncertainty when it comes to navigating civilian life. === Competence and the sense of capability === * Competence satisfaction refers to experiencing 'mastery', growth and effectiveness in skills. In the military this could translate to feeling effective and capable of mastering tactical, physical and technical demands. Frustration involves feeling ineffective, unable to succeed and incapable (Tillberg, et al. 2026). *Military training and their clearly defined roles may provide opportunities for feedback and achievement, which would help support competence. *However, transition may challenge these experiences when veterans encounter unfamiliar workplaces and/or struggle to communicate and translate their skills effectively. Study highlighted that a significant challenge for veterans' perceived competence was translating their military skills into civilian employment. However, successful job-search experiences, or discharging straight into a job may help restore/improve competence. Therefore, successful employment may have psychological influence beyond just financial security and could provide purpose, competence and structure (Tillberg, et al. 2026). === Relatedness and the need to belong === * Relatedness satisfaction involves mutual care, belonging and connections, whilst frustration can bring rejection, loneliness, alienation or exclusion. * Military service is known for providing strong sources of relatedness through unit cohesion, teamwork, shared military identity and experiences. Leaving the military has been found to disrupt established social networks and require veterans to develop new sources of connection within civilian communities (Barnett, et al. 2021). * Maintaining military connections may support some veterans in the transition, however, developing civilian social identities may be important for long-term adjustment. === Motivation and wellbeing === * The satisfaction of autonomy, competence and relatedness are essential for optimal functioning and wellbeing. The satisfaction of these needs provide positive organisational outcomes with a more positive attitude and less turnover intentions, stronger psychological understanding, and more positive health and overall well-being. Satisfaction of these needs also create a higher likelihood of intrinsic motivation (Linden, et al. 2026). * Autonomous motivation includes intrinsic motivation and positive internalised forms of extrinsic motivation, controlled motivation comprises of internal contingencies or external pressure. This distinction may be significant to veterans establishing new employment, social, educational and personal goals following discharge (Linden, et al. 2026). * ''Author note: maybe incorporate the 'why' behind their behaviour and how that is either positive or detrimental to motivation'' === Interacting needs === * Importantly, these three needs should not be considered completely independent as SDT conceptualises them as interrelated that collective contribute to optimal function and positive psychological wellbeing (Mobbs, et al. 2018). * Frustration in one area may also coincide with difficulties/challenges in another. * ''Author note: focus on research/studies examining the combinations and interactions between satisfaction/frustration rather than just an isolated predictor.'' == How does military experiences and reintegration shape autonomy, competence and relatedness? == * The psychological impact of military service can be shaped by the environment in which personnel operate, including the way that structure, occupational roles, leadership and social relationships are experienced (Barnett, et al. 2022). * The military service shouldn't be viewed as uniformly supportive or restrictive, it's important to factor which environmental conditions frustrate or facilitate psychological needs. * Reintegration changes these conditions irrespective of whether they were positive or negative environments, thus, requiring veterans to navigate new experiences, expectations, relationships and roles whilst renegotiating sources of autonomy, belonging and competence, see figure 2.[[File:Military to civilian transition.png|thumb|Figure 2: Environmental changes experienced during military-to-civilian reintegration]] * SDT therefore can be used to examine the conditions to which military experiences either support or undermine psychological need satisfaction, rather than assuming that military service is inherently beneficial or harmful. === Structure, hierarchy, and autonomy === * Military organisations rely heavily on rules, hierarchy, orders and clear responsibilities, shaping a high level of structure that is not often seen in civilian environments. * Structure is not inherently autonomy-thwarting as dependent as, structure can provide guidance, predictability, autonomy and clear expectations, especially, when personnel understand and endorse the purpose behind their responsibilities (Barnett, et al. 2021). * Leadership plays an integral role in satisfying autonomy (Knevelsrud, et al. 2023). In particular, autonomy-supportive leadership style is where leaders empower employees to make choices, share rationales, take perspectives and minimise unnecessary pressure. An autonomy-supportive leadership brings benefits of an higher engagement, positive wellbeing and enhanced performance (Knevelsrud, et al. 2023). * However, highly controlling environments may impede autonomy when their behaviour is experienced as pressured or coerced. This might be done through actions of micromanagement and excessive pressure, these may contribute to controlled motivation and poor psychological outcomes (Barnett, et al. 2021). * This creates an important distinction between control and structure, and that leadership style and individual interpretation. * Following either voluntary or involuntary discharge, veterans may experience substantially greater choice over relationships, future goals and employment, however, greater freedom does not necessarily equate to greater autonomy when personnel experience pressure, uncertainty or lack of meaningful alternatives. <quiz display=simple> {'''When can military structure support autonomy?''' |type="()"} + When personnel understand and value their purpose - When all choices are removed - When personnel experience coercion and external pressure - When responsibilities are increased {'''Which psychological need may be most affected when a veteran loses their military unit after discharge?''' |type="()"} - Competence + Relatedness - Belonging - Autonomy </quiz> === Training, occupational roles, and competence === * Military training can provide repeated opportunities and clearly defined occupation roles that enable feedback, skill development and achievement which supports competence. * Veterans may leave the service with diverse interpersonal and technical skills with a primary focus on leadership, management, communication, adaptability and teamwork. However, even though these skills can be transferable skills, it doesn't necessarily mean that veterans will experience or have their competence recognised within civilian employment and environments (Tillberg, et al. 2026). * A recurring challenge that veterans experience, is translating military-specific experience into skills that civilians will understand, this makes it challenging for veterans to confidently communicate the relevance of the capabilities. Difficulty translating skills can contribute to unsuitable employment, unemployment, despite veterans possessing valuable capabilities (Tillberg, et al. 2026). * This further highlights the important distinction between actual competence and perceived competence. Veterans may be highly capable whilst simultaneously experiencing uncertainty about their effectiveness in civilian environments. * Research from Becker, et al. 2022 involved 31 Aus Defence Force Members who had transitioned into civilian employment had identified challenges with career integration and adaptation, which further highlights the significance of organisational practices in helping veterans navigate employment and supporting longer-term retention. * From an SDT perspective, civilian employment may restore or strengthen competence through meaningful work and recognition, but could frustrate competence when their skills are overlooked or poorly aligned to available roles. * Successful vocational reintegration involves more that just securing employment, it about creating opportunities for veterans to recognise, demonstrate and develop their capabilities within their new occupational environment. === Relatedness: from military cohesion to civilian belonging === * Military service provides strong opportunities for relatedness through teamwork, unit cohesion, camaraderie, shared identity and goals. Consequently, belonging can become deeply embedded within the military environment which makes the loss of these social connections during the transition incredibly significant for their psychological adjustment (Barnett, et al. 2021). * SDT has demonstrated that shared experience and collective goals can shape a strong sense of social support and belonging. * Discharge can disrupt these established networks, which requires veterans to reconstruct social connection within civilian communities. * The relationship between belonging and military identity is incredibly complex, research from Flake and Kite 2021, used data from 358 Australian Defence Members and identified that identity change during the transition process can involve themes of loss, poor wellbeing, social isolation along with psychosocial transition challenges. This is in conjunction with sometimes an unwillingness to relinquish a dominant military identity can contribute to social disconnectedness which is seen through withdrawal or antisocial behaviour and while reinforcing a divide between military and civilian life. Subsequently, a disconnectedness from both the military and civilian environments increases the likelihood of a decreased effort to re-connect psychosocially which would support personal growth. Although, some aspects of military identity were associated with stronger connectedness and belonging (Barnett, et al. 2021). * A positive, protective trait that was found in research is that veterans with personal agency, pushing beyond existing boundaries and a willingness to evolve were identified as themes/traits that are associated with an effective transition (Barnett, et al. 2021). * This continues to reinforce the idea that successful reintegration may involve a veteran's source of belonging rather that just replacing their military community with a civilian one. * ''Key Point: the challenge is not just simply losing relationships, but losing an environment that embedded belonging into everyday life.'' === Identity and purpose === * The military structure is not inherently need-thwarting or need-supportive, the psychological impact depends on how rules, expectations and authority are experienced and communicated. * Structure can provide clear expectations and guidance, whereas, controlling practices may pressure individuals to behave in certain ways which consequently may undermine their sense of autonomy. * Military service is an important source of status, identity, purpose and social connection, especially, when an individual's role is clearly defined. * Leaving the military may involve loss of more than just employment and may experience uncertainty about their identity when their previous role is no longer a part of their everyday life. Research has revealed that loss of military identity, status and purpose as recurring aspects and challenges during the transition. * Veterans can go through identity renegotiation as they determine what aspects of military identity are meaningful and whether they would be useful in civilian settings. * Therefore, this complexity is important when considering the transition that military personnel experience as they leave an environment that was simultaneously structured and psychologically supportive but also restrictive in other ways. * ''Key highlight is whether military structure supports internalisation and psychological need satisfaction or becomes experienced as need thwarting and controlling.'' <br /> == How can Self Determination Theory informed interventions support psychological need satisfaction and successful integration? == * Understanding how psychological challenges of military-to-civilian transition provides a key basis for considering how transition support can promote rather than undermine psychological need satisfaction. * An SDT-informed approach reorients the focus from simply helping veterans cope with change to instead, creating social environments that support autonomy, competence and relatedness. * Therefore, existing transition support should be considered in terms of whether it provides opportunities for mastery, choice and meaningful social connection. === Creating supportive transition environments === * Reintegration support should address psychological as well as practical aspects of transition * SDT has suggested that environments can support adjustment when they support autonomy, competence and relatedness. * Transition planning has been found to support autonomy through meaningful choice, veteran-led goals, and collaborative decision-making. * Support should provide structure and guidance without becoming unnecessarily controlling, however, it's important to note that providing choices does not automatically satisfy autonomy. === Supporting competence and connection === * Vocational support can help veterans be able to translate military skills into civilian employment and also develop confidence in unfamiliar occupational environments and civilian environments. * Career counselling, feedback, recognition and retraining can strengthen perceived competence. * Services such as peer support, mentoring, family and community involvement can provide opportunities to develop and restore relatedness and belonging. * Effective support should enable veterans to develop new connections whilst allowing them to maintain meaningful military relationships, however, interventions should support multiple psychological needs simultaneously rather than separately. === Implications and limitations === * SDT can provide a framework for evaluating whether reintegration services create conditions that support rather than frustrate needs. * Psychological need satisfaction cannot explain reintegration of broader factors such as housing, finances and access to services. * Therefore, a important research gap is still evidence testing SDT-informed veteran interventions. More longitudinal studies and Australian-specific research should be conducted to determine whether supporting autonomy, competence and relatedness produces a sustained improve in wellbeing and reintegration. ;Embedded links (keeping for reference) * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia articles (e.g., "An early psychological view [[w:Dreams|dreams]]) of dreams was provided by [[w:Sigmund Freud|Sigmund Freud]]") or ** Related book chapters (e.g., "If you're feeling stuck, check out the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]") ==Conclusion== * Military-to-civilian reintegration is a highly complex psychological adjustment that involves changes to identity, daily structure, competence, purpose, relationships and roles. * SDT provides a useful framework to understand how these changes influence autonomy, competence and relatedness, along with the impact upon different types of motivation (autonomous and controlled motivation). * The key finding is that military experiences are not necessarily need-supportive or need-thwarting. Instead, dependent on how leadership, structure, responsibilities and relationships are experienced and understood. * Briefly discuss that identity and purpose are interlinked with the three psychological needs (autonomy, competence and relatedness). * SDT can inform practical interventions by creating transition environments that support identity, choice, competence, connection and meaningful goals. * Important take home message: success reintegration is not solely based on getting veterans into civilian life, instead, it should support veterans in developing self-directed, meaningful and socially connected lives beyond the military service. ==See also== * [[Social connection and emotion regulation|Social Connection]] (Book chapter, 2026)) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Only select links to major internal resources about the topic * Include the source in parentheses }} ==References== {{Hanging indent|1= Barnett, A., Savic, M., Lubman, D. (2021). Transitioning to civilian life: The importance of social group engagement and identity among Australian Defence Force veterans. ''Australian and New Zealand Journal of Psychiatry, 56''(8). https://doi.org/10.1177/00048674211046894 Becker, K., Bish, A., Abell, D., McCormack, M., Smidt, M. (2022). Supporting Australian veteran transition: career construction through a person-environment fit perspective. ''The International Journal of Human Resource Management, 36''(5), 799-823. https://doi.org/10.1080/09585192.2022.2077127 Carra, K., Curtin, M., Fortune, T., Gordon, B. (2022). Service and demographic factors, health, trauma exposure, and participation are associated with adjustment for former Australian Defence Force members. ''Military Psychology, 35''(5), 480-492. https://doi.org/10.1080/08995605.2022.2120312 Flack, M., Kite, L. (2021). Transition from military service to civilian: identity, social connectedness, and veteran wellbeing. ''PLoS ONE, 16''(12). https://doi.org/10.1371/journal.pone.0261634 Grimell, J. (2024). You can take a person out of the military, but you can't take the military out of the person: findings from a ten-year identity study on transition from military to civilian life. ''Frontiers in Sociology, 9.'' https://doi.org/10.3389/fsoc.2024.1406710 Kerr, N., Lane, S., Plotnikoff, R., Ashby, S. (2023). The "Transition" to civilian life from the perspective of former serving Australian Defence Force members. ''Journal of Veteran Studies, 9''(1), 129-142. https://10.21061/jvs.v9i1.407 {{ic|fix link}} Knevelsrud, H.C., Sorlie, H., Valaker, S. (2023). Mission command: A self determination theory perspective. ''Military Psychology, 36''(6), 672-688. https://doi.org/10.1080/08995605.2023.2252718 Linden, A., Borjesson, M. (2026). Motivation in physically demanding military roles: a qualitative study using self-determination theory. ''Military Psychology,'' 1-14. https://doi.org/10.1080/08995605.2026.2671587 Loverre, M., Chirico, A., Cinque, L., Palombi, T., Alivernini, F., Lucidi, F., Alessandri, G., Livi, S. (2024). A systematic review of self-determination theory's application in military and police organisations. ''Journal of Police and Criminal Psychology, 40''(5), 685-707. https://doi.org/10.1007/s11896-024-09718-2 Mobbs, M., Bonanno, G. (2018). Beyond war and PTSD: The crucial role of transition stress in the lives of military veterans. ''Clinical Psychology Review, 59,'' 137-144. https://doi.org/10.1016/j.cpr.2017.11.007 Olafsen, A., Halvari, H., Frolound, C. (2021). The basic psychological need satisfaction and need frustration at work scale: A validation study. ''Organisational Psychology, 12.'' https://doi.org/10.3389/fpsyg.2021.697306 Romaniuk, M., Saunders-Dow, E., Brown, K., Batterham, P. (2024). Feasibility, acceptability, and initial outcomes of a psychological adjustment and reintegration program for transitioned military veterans. ''BMC Psychology, 12''(1). https://doi.org/10.1186/s40359-024-02097-7 Ryan, R., Deci, E. (2022). Self determination theory. ''Encyclopedia of quality of life and well-being research.'' https://doi.org/10.1007/978-3-319-69909-7_2630-2 Rybakovaite. J., Poskus, M., Blue, S. (2021). Forced or free choice: Hardiness, need satisfaction, and engagement among military conscripts. ''Current Psychology, 42''(1), 7909-7919. https://doi.org/10.1007/s12144-021-02123-6 Tillberg, L., Truusa, T.T, Tillberg, P. (2026). The skillful veteran: Transforming overseas experience into competence, from the military to civilian working life domain. ''Vocations and Learning, 19''(5). https://doi.org/10.1007/s12186-026-09385-w }} APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sacks, O. (1985). ''The man who mistook his wife for a hat and other clinical tales''. Harper & Row. }} {{tip|Suggestions for this section: * Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]: ** Use "Edit source" ** <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> * Important aspects of APA referencing style ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case (except first letter and proper names), ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** doi as a URL which is a working hyperlink (i.e., clickable) * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** dois which aren't clickable as working hyperlinks ** Citing sources that haven't been consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as podcasts and videos, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]] and alphabetical order. For example: * [https://defenceveteransuicide.royalcommission.gov.au/publications/final-report Military-to-civilian Transition Final Report] (Royal Commission) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Self-determination theory]] [[Category:Motivation and emotion/Book/Veterans]] fotkxr30pe4tpbbbxqpawajdiz3rir7 0 331234 2835508 2835499 2026-10-02T13:04:35Z Mymunu 3106327 /* Social relationship and inclusion */ 2835508 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] 1p9k5pptp0852puvr95zmotvb4jp7dw 2835509 2835508 2026-10-02T13:07:29Z Mymunu 3106327 /* References */ 2835509 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not claim a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes being measured, the timing of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence. In contrast, other findings differed according to the outcome, timing and context being examined. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings. Lee et al. (2025) also found different associations according to the timing of outdoor play. Dodd et al. (2026) provides longer-term evidence. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood. However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] 5c5ceb94i7er1oshbmtiso7sd00zxi7 2835606 2835509 2026-10-03T04:33:17Z Mymunu 3106327 /* What does recent research show about outdoor play and children's emotional well-being? */ 2835606 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2024) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. This differences shows that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from the studies done by Poulos et al. (2026), which examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being are not same across outcomes.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] ag1dv173emhxln9oxfin3r0bqmrvhts 2835607 2835606 2026-10-03T04:40:08Z Mymunu 3106327 /* Consistency of recent research findings */ 2835607 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence, including lower difficulties and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2025) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. These differences show that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from Poulos et al. (2026), who examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being is not same across outcome.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from r4,151 children and measured outdoor play during the preschool years approximately 2 to 4 years old, and internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] 97uf77bi9w886nbx86ky0btniyovhfx 2835608 2835607 2026-10-03T04:44:21Z Mymunu 3106327 /* Strengths and limitations of current research */ 2835608 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence, including lower difficulties and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2025) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. These differences show that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from Poulos et al. (2026), who examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being is not same across outcome.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from 4,151 children and measured outdoor play during the preschool years at approximately 2, 3 and 4 years old, while internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2024) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being remains less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2024) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Relied on parent reported measures, and the observational data cannot not establish conclusions. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared positive and negative affect after outdoor gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in table 3, the recent evidence has different methodological strengths and limitations. Longitudinal design provide stronger evidence about temporal ordering than cross-sectional design studies, while device based measured can address some limitations of parent report. However, much of the evidence remains observational, differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] b3mpyhmsqov7aityk0qt9rgkunc9iqb 2835613 2835608 2026-10-03T04:52:59Z Mymunu 3106327 /* Strengths and limitations of current research */ 2835613 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence, including lower difficulties and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2025) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. These differences show that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from Poulos et al. (2026), who examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being is not same across outcome.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from 4,151 children and measured outdoor play during the preschool years at approximately 2, 3 and 4 years old, while internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2025) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small (N=107) and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being is less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; the study also considered possible reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2025) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Outdoor play and symptoms relied on parent reported measures, and the observational design does not establish causations. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared children's positive and negative affect following outdoor, gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in Table 3, the recent evidence has different methodological strengths and limitations. Longitudinal designs provide stronger evidence about temporal ordering than cross-sectional design studies, while device-based measures can address some limitations of parent report. However, much of the evidence remains observational, and differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|figure 5]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. This provides a broader context for considering the roles of families, communities and schools. === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematically reviewed 21 peer reviewed studies of children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents attitude towards outdoor play may shape children's opportunities to play outdoor. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support also requires to allow for children for children's independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ how much independence they prefer (Han et al., 2022). The study identified different patterns in children and parents preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting for greater independence. Beekhuizen et al. (2025) also recommended providing children and parents knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research provides more evidence about what happens when children play together. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The findings reviewed to show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socioemotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. * Schools can provide opportunities for children with and without disabilities to become familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. * School circumstances can also affect the time and opportunities available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing the time available to play outdoors. Beekhuizen et al. (2025) identified these experiences under time constraints. * Recess provides another school setting in which children's emotional experiences have been examined. Poulos et al. (2026) studied 317 year 4 and 5 children during extreme heat weather and found significantly higher positive affect after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three recess settings. * Poulos et al. (2026) identified implications for school health, policy and equity, including the need for heat adaptive infrastructure to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher following gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Children who have access to the same outdoor playground may still have different opportunities to participate in play. |type="()"} + True - False </quiz> {{Robelbox/close}} {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Outdoor play has the same relationship with children's emotional well-being regardless of how door play or emotional well being is measured. |type="()"} - True + False </quiz> {{Robelbox/close}} {| class="wikitable" |Scenario Two primary school aged children A and B enjoy playing outdoors. They both lived near the same playground. Child A was very familiar with the playground and knows several other children who play there and easily join the games. While child B has disability, uses mobility device and does not know other children at playground. Some parts of the playground are difficult to access and joining other children who are already playing can also be challenging. Child B's parents can provide support, although the child also prefers to play more independently. Consider both children can access the same playground, but do they have same level opportunities to take part in outdoor play? |} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] potw26opcx9zlr2avnssakzdwo2v1p0 2835630 2835613 2026-10-03T06:00:09Z Mymunu 3106327 /* How do families, communities and schools support outdoor play opportunities? */ 2835630 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence, including lower difficulties and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2025) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. These differences show that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from Poulos et al. (2026), who examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being is not same across outcome.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from 4,151 children and measured outdoor play during the preschool years at approximately 2, 3 and 4 years old, while internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2025) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small (N=107) and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being is less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; the study also considered possible reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2025) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Outdoor play and symptoms relied on parent reported measures, and the observational design does not establish causations. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared children's positive and negative affect following outdoor, gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in Table 3, the recent evidence has different methodological strengths and limitations. Longitudinal designs provide stronger evidence about temporal ordering than cross-sectional design studies, while device-based measures can address some limitations of parent report. However, much of the evidence remains observational, and differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|Figure 5. Families, communities and school can shape children's opportunities for play.]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. Together, these findings provide a broader context for considering the roles of families, communities and schools. These influences can operate across family, community and school settings (see Figure 5). === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematic review of 21 peer reviewed studies involving children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents' attitude towards outdoor play may shape children's opportunities to play outdoors. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support can also involve allowing children to have some independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ in how much independence they prefer (Han et al., 2022). The study identified different patterns in children's and parents' preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting greater independence. Beekhuizen et al. (2025) also recommended providing children and parents with knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The reviewed findings show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socio-emotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Which statement best reflects how families, communities and school can support children's outdoor play opportunities? |type="()"} + Support can involve accessible environments, safety, familiarity, appropriate parental support and opportunities for children to participate with some independence. - Providing the same outdoor space gives every child the same opportunity to participate. -Schoolyard greening alone is enough to improve children's emotional well-being. -Outdoor recess always produces better emotional outcomes than indoor recess. </quiz> {{Robelbox/close}} == '''Conclusion''' == * Outdoor play and children's emotional well-being are measured in different ways across recent research. Outdoor play is not equivalent to outdoor physical activity, while emotional outcomes include emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences need to be considered when findings across studies are compared. * Research also provides mixed evidence about how outdoor play may be related to emotional well-being. Gross's (2015) emotion-regulation framework can be applied to emotional situations experienced during play, but it does not show that outdoor play improves emotion regulation. Lee et al. (2025) provides more direct evidence of a possible cognitive process, with working memory statistically mediating the association between afternoon outdoor play and lower emotional dysregulation. However, this observational finding does not establish causation. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Recent findings are not consistent across all outcomes or contexts. Ferguson et al. (2025) reported associations with better social-emotional competence, Dodd et al. (2026) found longitudinal associations with internalising and externalising symptom trajectories, and Poulos et al. (2026) found differences in positive but not negative affect following recess. In contrast, most associations examined by Davenport et al. (2024) were not statistically significant. * Overall, current research supports an association between outdoor play and some aspects of children's emotional well-being but does not establish one consistent effect. Differences in measurement, research design, children and play contexts remain important when interpreting the evidence. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] c44wcoghrwxojg0eiklxxzqo6gjlwlx 2835633 2835630 2026-10-03T06:16:42Z Mymunu 3106327 /* Conclusion */ 2835633 wikitext text/x-wiki {{title|Outdoor play and children's emotional well-being:<br>How does outdoor play influence children's emotional well-being?}} __TOC__ == Overview == {| class="wikitable" |'''Case study: Part 1'''[[File:School children happily playing in playground.jpg|thumb|'''Figure 1'''. Children engaged in outdoor play, where social interactions, choice and control on the situation can create different emotional experiences.]]Imagine that... A child arrives at playground and joins a group of children who are already playing. The children are making their own game and deciding the rules as they play. At first, the child enjoys choosing where and what to play. During the game, another child changes the rules of the game and argument breaks out. The child becomes frustrated and walks away from the activity. After watching for a short time, the child returns, suggests a new idea to continue the game and re-joins the group. How might experiences like this during outdoor play influence children's emotional well-being? Emotional regulation provides one psychological framework for understanding how children may respond to situations like this during play (see figure 1). |} Outdoor play can involve more than physical movement. It can occur across playgrounds, gardens, streets and natural spaces. Outdoor play includes exploration, social interaction, choice, challenge, and engagement with the surrounding environment (Dodd et al., 2026; Pereira et al., 2024). As shown in the case study part 1, outdoor play can involve emotional situations such as [[Aggression|frustration]], disagreement, uncertainty or excitement. Gross's (2015) [[emotional regulation]] provides a [[psychological]] explanation for how children may recognise and respond to these emotional situations. Outdoor play experiences are also shaped by the environment where play takes place. Different physical and social environments can provide different opportunities for play, which means the same outdoor space may not provide the same opportunities for every child (Morgenthaler et al., 2024).This chapter examines how [[Emotional regulation|emotional,]] [[Cognitive psychology|cognitive]], social and environmental processes may help explain the relationship between outdoor play and children's [https://w.wiki/AHG6 emotional well-being.] {| class="wikitable" | ;[[File:Bulb Idea Flat Icon GIF Animation.gif|alt=Focus questions|left|99x99px|thumb]] '''Focus questions''' * '''What are outdoor play and children's emotional well-being?''' * '''How does outdoor play influence children's emotional well-being?''' * '''How do social and environmental conditions shape children's outdoor play experiences?''' * '''What does recent research show about outdoor play and children's emotional well-being?''' * '''How can families, communities and schools support outdoor play opportunities that may benefit children's emotional well-being?''' |} == '''Understanding outdoor play and children's emotional well-being''' == [[File:Water-outdoor-people-girl-play-boy-938233.jpg|thumb|'''Figure 2'''. ''Outdoor play can provide opportunities to explore, make choices, interact with others and respond to changing situations through activities such as water play.''|214x214px]] Outdoor play and children's emotional well-being are not measured as single and consistent concepts across recent research. Outdoor play may be examined as voluntary and intrinsically motivated activity, self directed play, parent reported outdoor play, or device-measured outdoor play (Binno et al., 2025; Davenport et al., 2025; Dodd et al., 2026). It should also be distinguished from outdoor moderate-to-vigorous physical activity (MVPA) because these measures do not represent the same behaviour (Davenport et al., 2025). Children's emotional well-being is also examined through different outcomes, which include [https://w.wiki/7YC8 emotional dysregulation], social-emotional competence, positive and negative affect, and internalising or externalising symptoms (Dodd et al., 2026; Ferguson et al., 2025; Lee et al., 2025; Poulos et al., 2026). These differences need to be considered when findings across the studies are compared. === Outdoor play === Outdoor play is voluntary and and intrinsically motivated activity that occurs in outdoor environments such as gardens, playgrounds, parks and natural spaces (Dodd et al., 2026). Free play can be also self-directed, allowing children to interact freely with the environment and have [https://w.wiki/4NyU autonomy] to make choices about their activities (Biino et al., 2025). Outdoor play should not be treated as the same as outdoor physical activity. The recent research outdoor play and outdoor moderate- to-vigorous physical activity (MVPA) were examined as separate measures (Davenport et al., 2025). This distinction is important because outdoor play intensity of physical activity do not represent the same behaviour. === Children's emotional well-being === Children's emotional well-being is not represented by a single common outcome across recent outdoor play research. Emotional dysregulation has been examined in relation to the amount and timing of outdoor play (Lee et al., 2025), while social-emotional competence includes social emotional difficulties and [[Prosocial behavior/Keywords/Definitions|prosocial behaviour]] (Ferguson et al., 2025). Other studies have examined positive and negative affect following recess (Poulos et al., 2026) and trajectories of internalising and externalising symptoms across childhood (Dodd et al., 2026). These outcomes represent different aspects and timeframes of emotional functioning, so they should not be treated as equivalent when findings are compared. '''Table 1''' ''Emotional outcomes examined in recent research on outdoor play and children's emotional well-being'' {| class="wikitable" |+ !Study !Outcome examined !What the outcome represents |- |Lee et al. (2025) |Emotional dysregulation |Difficulties in emotional regulation; examined in relation to the amount of timing of outdoor play |- |Ferguson et al. (2025) |Social-emotional competence |Social-emotional difficulties and prosocial behaviour |- |Poulos et al. (2026) |Positive and negative affect |Children's affect following recess across different settings |- |Dodd et al. (2026) |Internalising and externalising symptoms |Internalising: emotional symptoms and peer problems Externalising: conduct problems and hyperactivity- inattention |} Note. Outcome descriptions are based on the measures and constructs reported in the above cited studies. See Table 1. As shown in Table 1, the studies do not measure the same aspect or time frame of children's emotional functioning. Affect immediately following recess should not be treated as equivalent to emotional dysregulation measured over time or trajectories of internalising and externalising symptoms across childhood. This difference need to be considered when findings across studies are compared. '''Case study: part 2''' ''What is being measured?'' {| class="wikitable" | valign="top" |[[File:Children marbles.jpg|thumb|221x221px|'''Figure 3'''. ''Children's immediate emotional responses during outdoor play is different from long-term emotional functioning.'']]The child in the Case study: Part 1 becomes frustrated when the rules of the game change. This immediate emotional response is different from a longer-term pattern of emotional dysregulation or internalising symptoms. Research on outdoor play measures these outcomes across different time frames and contexts. When findings are compared, it is important to consider which aspect of emotional well-being is measured. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {A child's positive or negative affect immediately after recess represents the same emotional outcome as longer-term internalising or externalising symptoms. |type="()"} - True + False </quiz> {{Robelbox/close}} |} == '''How does outdoor play influence children's emotional well-being?''' == Outdoor play may be related to children's emotional well-being through emotional, cognitive, and social processes (Dodd et al., 2026). Possible pathways discussed in the recent research include cognitive, social and emotional skills, social interaction, physical activity, connection with the nature and adventurous play (Dodd et al., 2026). However, these pathways have not all been directly tested as mechanisms, and many of the outdoor play research remain observational (Dodd et al., 2026; de Lannoy et al., 2023). Psychological theory can help to explain how and why outdoor play experiences may influence children's emotional well-being. Although the evidence does not yet show that psychological processes cause the relationship between outdoor play and children's emotional well-being. Emotion involves more than subjective feelings. Scherer (2005) described emotion as coordinated changes across cognitive appraisal, physiological responses, action tendencies, expression and subjective feelings. Therefore, physiological responses are one component of an emotional response rather than a separate explanation for the effects of outdoor play. Scherer (2005) did not examine outdoor play, so this framework does not provide evidence that outdoor play changes children's physiological responses or emotional well-being. === '''Emotion regulation''' === Emotion regulation provides a psychological framework for understanding how children may manage emotional situations during outdoor play. Gross (2015) described emotion regulation as a process involving identification, selection and implementation. Identification concerns deciding whether to regulate an emotion, selection concerns selecting regulation strategy and implementation concerns applying chosen response into practice. Gross (2015) also described situation selection, situation modification, attentional deployment cognitive change and response modulation. During outdoor play, children may experience frustration, conflict, disagreement, uncertainty or changes to their activity. Gross's framework can be applied to these situations to consider how the children recognise an emotional response, choose strategies to manage and act on chosen strategies. However, Gross (2015) did not examine outdoor play, so applying the framework does not show that outdoor play improves children's emotion regulation. Broader research on children's play has also examined emotional functioning. Veraksa et al. (2025) systematically reviewed 33 studies on non-therapeutic play among children aged 3 to 7 and identified evidence involving emotional regulation, emotional intelligence, and negative emotional symptoms. The review included outdoor play alongside several other forms of play, so its findings should not be interpreted as evidence that outdoor play alone improves emotion regulation. {| class="wikitable" | valign="top" |'''Case study: Part 3'''[[File:14Y Ncdc july 5th-802.jpg|thumb|202x202px|'''Figure 4'''. ''Children can respond differently to emotional situations that occur during outdoor play.'']]When the rules of game change, the child becomes frustrated and decides to step away. After a short time the child returns and suggests a different way to continue the game (see Figure 4). Suggesting a change to the game can be considered an example of situation modification. This shows how Gross's emotion regulation framework can be applied to an emotional situation during outdoor play. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {What does Gross (2015) emotion regulation framework show when it is applied to the outdoor-play scenario?. |type="()"} -A. Outdoor play causes better emotion regulation. +B. Situation modification can help explain how the child responds to the situation. -C. Children regulate emotions better outdoors than indoors. -D. Stepping away from play always improves emotional well-being. </quiz> {{Robelbox/close}} |} === '''Working memory and cognitive process''' === Working memory can be one cognitive process involved in the association between outdoor play and emotional dysregulation. Lee et al. (2025) examined 325 preschool children and tested working memory as a statistical mediator. Outdoor play was examined across three periods of the day; wake up to noon, noon to 6 pm, and 6 pm to bedtime. The findings differed according to the timing of outdoor play. More outdoor play between noon and 6 pm was associated with lower emotional dysregulation. In contrast, more outdoor play before noon and after 6 pm was associated with higher emotional dysregulation. Afternoon outdoor play was also associated with better working memory, and better working memory was associated with lower emotional dysregulation. When working memory was included in the analysis, the association between afternoon outdoor play and emotional dysregulation was no longer statistically significant. Lee et al. (2025) reported this as full mediation. This suggests that working memory may be involved in the association between afternoon outdoor play and lower emotional dysregulation. However, the study was observational, so the mediation analysis does not show that outdoor play caused better working memory or lower emotional dysregulation. Johnstone et al. (2022) reviewed 36 studies of nature-based [[early childhood education]]. The review found very low certainty evidence for some social, emotional, and cognitive outcomes, including [[self-regulation]], while findings for attention were inconsistent. Although, the review examined nature-based early childhood education rather than outdoor play. Therefore, it does not show that working memory explains the association between outdoor play and children's [https://w.wiki/AHG6 emotional well-being]. == '''How do social and environmental conditions influence on children's outdoor play experiences?''' == Outdoor play opportunities depend on social and physical environments. Children may have access to same outdoor space but may not have same opportunities for play. Beekhuizen et al. (2025) identified personal, interacting and environmental factors that could support or restrict inclusive outdoor play. These difference can be considered through [https://w.wiki/VYfL affordance theory], which examines the opportunities for action through the relationship between an individual and the environment (Gibson, 1979). === Affordance theory and outdoor play environments === [https://w.wiki/VYfL Affordance theory] was developed by Gibson (1979) to explain the opportunities for action that an environment provides in relation to the person using it. In outdoor play, the opportunities provided by a space can depend on its physical features and how a child is able to use those features. Morgenthaler et al. (2024) applied affordance theory to children's outdoor play and examined outdoor play transactions through environmental taxonomy of outdoor play space features. Their review identified 14 categories of spaces and 284 play opportunities linked to different environmental features. Morgenthaler et al. (2024) showed that different spaces and objects can afford different forms of play, which means two children using the same outdoor play space may not experience the same play opportunities. Therefore, the theory provides a way of understanding differences between children and their use of an outdoor environment. However this study examined the opportunities for play rather than children's well-being. Pereira et al. (2024) also examined features of outdoor play spaces in a systematic review of 51 articles representing 45 primary studies. The review examined features such as fixed structures, natural elements, floor markings, loose equipment and available play areas. Some positive effects were reported, but there was substantial heterogeneity between studies. The studies differed in the environmental features examined, outcomes measures, and study methods, which made difficult to draw a clear conclusions about the effect of individual play space features (Pereira et al., 2024).Therefore, the review does not show that one particular environmental feature improves children's [https://w.wiki/AHG6 emotional well-being]. '''Table 2'''. ''Examples of environmental features and possible affordances'' {| class="wikitable" |+ !Environmental feature !Possible affordance for play |- |Fixed structures (e.g. climbing frame) |Can provide opportunity in different forms of physical activity. For example, climbing, physical play, risk-taking |- |Natural elements (e.g. tree, rocks, sand) |Environmental features can provide opportunities for exploratory play, imaginative play, sensory experiences |- |Floor markings (e.g. painted games) |Can support structured activities like rule based play, footy, tennis, etc. |- |Loose equipment (e.g. balls, tyre, boxes) |Movable environmental features that can be used during constructive play, imaginative play, creative problem solving play |- |Open play area |Running, group play, or flexible use |} Note. This table is an original synthesis based on environmental features described by Pereira et al. (2024) and affordance framework applied by Morgenthaler et al. (2024). Environmental features can afford more than one type of play and the opportunities may vary between children. {| class="wikitable" | valign="top" |'''Case study: Part 4 Same space, different opportunities''' Kim and Leo are both at the same school playground during recess. The playground has a climbing structure, a sand pit area and open natural space. * Kim quickly joins a group of children using a climbing frame. The climbing frame structure provides opportunities for physical play and social interactions. * Leo uses a sand pit with other child. He plays with loose materials and creates a sand castle and other different structures. The environment gives him an opportunity for imaginative play and social play. These examples show how the same outdoor space can afford different forms of play for different children, depending on their interests, skills and social context. {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Which statement best applies affordance theory to the example?. |type="()"} -A. The playground causes children to be more physically active. +B. The same playground can provide different opportunities for play depending on the child and how they use the environment. -C. Climbing structures always lead to better emotional well-being. -D. Open spaces are only useful for social play. </quiz> {{Robelbox/close}} |} === '''Social relationship and inclusion''' === Social relationships can shape whether children are able to join and continue outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified making contact and growing up together as interacting factors in inclusive outdoor play. Familiarity with other children could make it easier to approach, communicate and play together, while joining unfamiliar children or existing game could be difficult. [https://w.wiki/WDsJ Affective social competence (ASC)] provides a psychological framework for understanding these social interactions (Halberstadt et al., 2001). The ASC model involves experiencing, sending and receiving affect, together with awareness, identification, working within the social context and regulation. These processes may be relevant when a child tries to enter a peer group, recognise other children's emotions and manage their own response during play. Gal-Szabo et al. (2018) used the ASC model as theoretical foundation when examining children's emotional knowledge and social behaviour during preschool free play. The study found that situational understanding of the emotions was negatively related to uninvolved behaviour and was marginally positively related to more social play. However, the study examined preschool free play rather than specific outdoor play. Beekhuizen et al. (2025) also found that children with disabilities experienced more barriers related to making contact, adapting play and physical play environment. These findings suggest that participation in outdoor play depends on both social interaction and children's ability to respond within the play situation, but they do not establish that social play causes better emotional well-being (Beekhuizen et al., 2025; Gal-Szabo et al., 2018). === '''Autonomy, parents and time''' === [https://w.wiki/4NyU Autonomy] can shape how children experience outdoor play (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified autonomy as a personal factor in inclusive outdoor play. Some children valued having independence over what they do and who they play with. Some children with disabilities wanted opportunities to play outside independently without direct parental involvement. Autonomy during play may also have an emotional function. Veraksa et al. (2025), in a systematic review of non-therapeutic play, identified autonomy as one feature through which play may support emotional regulation and coping. However, the review examined several forms of play and does not show that autonomy during outdoor play specially improves children's emotional well-being. Parental support can also make outdoor play possible. Beekhuizen et al. (2025) found that parental involvement could support children's participation when a child needed help to access or participate in outdoor play. Time constraints also affected opportunities for outdoor play. Some children had less time to play outdoors because of parents' work schedule, after school arrangements or longer travel distance between home and school (Beekhuizen et al., 2025). These constraints could limit opportunities for outdoor play even when children wanted greater independence. === '''Outdoor environments and accessibility''' === Outdoor play environments can support or restrict children's participation (Beekhuizen et al., 2025). Beekhuizen et al. (2025) identified both physical and social features of outdoor environments as relevant to inclusive play. Children discussed safety, including traffic near playgrounds and disputes with other children. Accessibility was particularly important for children with disabilities because opportunities for inclusive outdoor play could be affected by play environments and physical abilities. Pereira et al. (2024) also showed that heterogeneity between studies limited clear conclusions about the effects of individual outdoor play space features.. Gibson et al. (2017) systematically reviewed five studies of loose-parts play (LPP) interventions with primary school aged children. The review found limited quantitative evidence to determine whether loose-parts interventions affected children's cognitive, social or emotional development. Together, these findings show that features of outdoor environments can support or restrict opportunities for play. However, current evidence does not establish that a particular environmental feature that improves children's emotional well-being (Beekhuizen et al., 2025; Gibson et al., 2017; Pereira et al., 2024). {| class="wikitable" | valign="top" |'''Case study: Part 5 Same play space, different experiences''' {| class="wikitable" |Kim and Leo use the same school playground. Kim already knows many children and easily joins a group activity. Leo uses a mobility device and finds that some areas of the playground are difficult to access. Leo also does not know many children who are already playing. He prefers to play independently when possible. |} The two children have access to the same playground space, but their opportunities to participate are different. There experiences are shaped by more than the physical space alone. Beekhuizen et al. (2025) also identified making contact, familiarity, autonomy and accessibility as factors that could support or restrict inclusive outdoor play. Physical accessibility may affect whether a child can use part of playground, while familiarity with other children may affect whether the child can enter social play. Parental support may also help participation in some situations, while the child may still value greater independence {{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why might two children have different outdoor-play experiences even they use the same playground?. |type="()"} +A. Physical, social and personal factors can affect how children are able to participate. -B. Outdoor spaces provide the same opportunities for every child. -C. Accessibility is the only factor that affects outdoor play.. -D. Children who know other children always have better emotional well-being. </quiz> {{Robelbox/close}} |} == '''What does recent research show about outdoor play and children's emotional well-being?''' == Recent research does not show a consistent association between outdoor play and children's emotional well-being. The findings are mixed across different emotional outcomes measured, the timing and context of outdoor play, children's behaviours, and the methods used to measure outdoor play. These differences are significant when interpreting whether outdoor play is associated with better emotional well-being. The recent studies shows this variation. More frequent outdoor play has been associated with better social-emotional competence (Ferguson et al., 2025) while association with emotional dysregulation differed according to the timing and context of outdoor play (Lee et al., 2025). Positive affect was also higher after outdoor play and and gymnasium recess than after classroom recess, but no significant differences in negative affect between settings (Poulos et al.,2026). Longer-term evidence also examines a different aspect of emotional functioning. More frequent outdoor play during the preschool years was associated with a greater likelihood of following a low and stable trajectory of internalising and externalising symptoms through childhood (Dodd et al., 2026). However, the studies differ in children's ages, research designs, measures of outdoor play and emotional outcomes. Therefore, the evidence should not be interpreted as showing one consistent effect across all children or forms of emotional well-being. === '''Consistency of recent research findings''' === Recent findings are not consistent across all emotional and developmental outcomes. Ferguson et al. (2025) found that more frequent outdoor play was associated with better social-emotional competence, including lower difficulties and greater [[Prosocial Behaviour|prosocial behaviour]]. In contrast, Davenport et al. (2025) found that most associations between outdoor play and the physical, [[Cognitive psychology|cognitive]] and social-emotional developmental indicators examined were not statistically significant. These differences show that positive association with outdoor play are not found across all developmental outcomes. Findings can also differ according to the emotional outcome being measured. Poulos et al. (2026) found higher positive affect after outdoor and gymnasium recess than after classroom recess, but no significant differences in negative affect between the three recess settings. Lee et al. (2025) found that more afternoon outdoor play was associated with lower emotional dysregulation, while more outdoor play earlier and later in the day was associated with higher emotional dysregulation. These findings suggest that the association can differ according to both the emotional outcome and the timing of outdoor play. The time period being examined also differs between studies. Dodd et al. (2026) found that more frequent outdoor play during the preschool years was associated with lower odds of belonging to less favourable internalising and externalising symptom trajectory across childhood. This is different from Poulos et al. (2026), who examined children's emotional states immediately after the recess. Therefore, recent research includes both immediate emotional outcomes and longer-term patterns of emotional functioning. Associations may vary across children and play contexts. Ferguson et al. (2025) reported differences according to ethnicity and whether outdoor play occurred on school days or weekends. Together, these findings suggest that the association between outdoor play and emotional well-being is not same across outcome.The studies differ in children's age, research designs, measures of outdoor play, and emotional outcomes. Thus, findings should not be treated as evidence of one single effect. === '''Strengths and limitations of current research''' === The strength of the evidence also differs according to research design. Dodd et al. (2026) used longitudinal data from 4,151 children and measured outdoor play during the preschool years at approximately 2, 3 and 4 years old, while internalising and externalising symptoms trajectories were measured later through to age 8. This strengthens the evidence about temporal ordering because outdoor play was measured before the later emotional outcomes. However, longitudinal evidence does not establish cause and effect, and the study relied on parent-reported measures. Causal interpretation is also limited in other studies. Lee et al. (2025) used observational data and considered the possibility of reverse causation when interpreting associations between the timing of outdoor play and emotional dysregulation. This means the findings cannot establish whether outdoor play influenced emotional dysregulation or whether children's emotional functioning influenced when they played outdoors. Measurement of outdoor play is another important consideration. Davenport et al. (2025) used both parent-reported and device-measured outdoor play and examined whether associations were independent of outdoor moderate-to-vigorous physical activity (MVPA). This is important because outdoor play and outdoor physical activity are related but should not be treated as interchangeable measures. However, the sample was relatively small (N=107) and most tested associations with developmental outcomes were not statistically significant. Limitations are also present across the wider outdoor-play literature. De Lannoy et al. (2023) identified 275 Canadian outdoor-play publications and found that cross-sectional designs were the most common. Mental and emotional development had also received less research attention than physical health and development. This suggests that the evidence base for emotional well-being is less developed than some other areas of outdoor-play research. T'''able 3. ''Comparison of recent research''''' {| class="wikitable" |+ !Study !Research method !Research strength !Limitation |- | <small>Lee et al. (2025)</small> |Observational; 6 month follow-up |Examined the timing of outdoor play and tested working memory as a statistical mediator of the association with emotional dysregulation. |Observational mediation cannot establish causation; the study also considered possible reverse causation when interpreting timing-related findings. |- |Ferguson et al. (2025) |Cross sectional; multi ethnic urban cohort (N=2,568) |Large sample allowed associations to be examined across population groups and school-day/weekend outdoor play. |Cross-sectional design cannot establish temporal direction; associations also differed across some groups and play contexts. |- |Davenport et al. (2025) |Observational; preschool sample (N=107) |Used both parent-reported and device-measured outdoor play and distinguished outdoor play from outdoor moderate-to-vigorous physical activity (MVPA) |Small sample and most tested associations with developmental outcomes were not statistically significant. |- |Dodd et al. (2026) |Longitudinal cohort (N = 4,151) |Outdoor play was measured in the preschool years before later internalising and externalising symptom trajectories. |Outdoor play and symptoms relied on parent reported measures, and the observational design does not establish causations. |- |Poulos et al. (2026) |Recess comparison study (N=317) |Compared children's positive and negative affect following outdoor, gymnasium and classroom recess |Examined immediate affect during an extreme-heat context; higher positive affect was also found after gymnasium recess, so the result was not specific to outdoor recess. |} As shown in Table 3, the recent evidence has different methodological strengths and limitations. Longitudinal designs provide stronger evidence about temporal ordering than cross-sectional design studies, while device-based measures can address some limitations of parent report. However, much of the evidence remains observational, and differences in emotional outcomes, measures of outdoor play and play contexts limit the direct comparison between studies. Future research could benefit from more consistent measures of outdoor play, stronger longitudinal and experimental designs and greater attention to emotional outcomes across different children and play contexts. {| class="wikitable" | valign="top" |{{Robelbox|theme=12|title=Test your learning}} <quiz display=simple> {Why should the recent findings not be interpreted as showing single consistent effect of outdoor play on children's emotional well-being? |type="()"} -A. Because all the studies used different countries. +B. Because studies differed in emotional outcomes, measures of outdoor play, timing, context and research design. -C. Because longitudinal studies are always more reliable and accurate. -D. Because outdoor play only affects positive outcomes. </quiz> {{Robelbox/close}} |} == '''How do families, communities and schools support outdoor play opportunities?''' == [[File:Coalition Builds New Playground in One Day for Chicora-Cherokee Community (11054532905).jpg|thumb|325x325px|Figure 5. Families, communities and school can shape children's opportunities for play.]] Outdoor play opportunities are shaped by children's social and physical environments. Beekhuizen et al. (2025) identified parents, outdoor play environments, communities and time constraints as environmental factors in inclusive outdoor play. The study also identified autonomy, growing up together, making contact and adapting ways of playing as factors that affected participation. The 2025 Position Statement on Active Outdoor Play recommends increasing opportunities for active outdoor play where people live, learn, work and play (Lee et al., 2025). It also recommends collaboration across sectors and settings to support equitable access. Together, these findings provide a broader context for considering the roles of families, communities and schools. These influences can operate across family, community and school settings (see Figure 5). === '''Families''' === Parents can influence whether children have opportunities to play outdoors. Beekhuizen et al. (2025) identified parents as an environmental factor in inclusive outdoor play. Children with disabilities described parents as important in encouraging participation. For some children, parents also helped to explain their disability to other children (Beekhuizen et al., 2025).   Broader research also identifies parental factors in children's outdoor play. A systematic review of 21 peer reviewed studies involving children aged 0 to 12 years found that the importance placed on outdoor play was associated with how much time children spent playing outdoors (Boxberger and Reimers, 2019). Evidence for parental encouragement and support was less consistent across children (Boxberger and Reimers, 2019). This suggests that family influence is not limited to direct encouragement. Parents' attitude towards outdoor play may shape children's opportunities to play outdoors. However, the review identified some associations but does not show that parental attitude caused greater outdoor play. Parental support can also involve allowing children to have some independence during outdoor play. Support can make participation possible, but it does not always require direct parental involvement (Beekhuizen et al., 2025). This is particularly relevant when children want more control over how and where they play. Research on independent mobility also shows that children and parents may differ in how much independence they prefer (Han et al., 2022). The study identified different patterns in children's and parents' preferences for independent mobility. These findings show that parental support and autonomy do not always involve the same level of parental involvement. Some children may need support to access or join outdoor play while also wanting greater independence. Beekhuizen et al. (2025) also recommended providing children and parents with knowledge about disabilities and possible adaptations that could support participation. Time constraints can also limit family-level opportunities for outdoor play. Beekhuizen et al. (2025) reported that some children had less time to play outdoors because of parents' work schedules, after-school arrangements and longer travel between home and school. This means that opportunities for outdoor play can be restricted even when children want to play and parents are supportive. === '''Communities''' === Community conditions can affect whether children have safe and inclusive opportunities for outdoor play. Beekhuizen et al. (2025) described physical and social safety as important for inclusive outdoor play. Some playground structures also made participation difficult for children with disabilities. A systematic review of 25 studies found that both physical and social neighbourhood factors were associated with children's outdoor play (Visser & van Aalst, 2022). Social factors included perceived safety, social cohesion and the presence of other children. However, the studies did not identify one neighbourhood feature that consistently explained children's outdoor play across all settings. The social environment may also affect whether children can join others. Children with disabilities described smaller communities, including villages and schools, as places where children and parents could become more familiar with disability. Greater familiarity was associated with greater acceptance and stronger connections with other children (Beekhuizen et al., 2025). Observational research also shows that physical access alone does not explain inclusive outdoor play. Beekhuizen et al. (2026) observed 63 children with and without disabilities during six inclusive outdoor-play sessions and found children with disabilities had lower playfulness scores than children without disabilities. Unfamiliarity could also restrict social inclusion because children preferred to play with peers they already knew. This means providing the same outdoor space does not necessarily provide the same opportunity to participate. The 2025 Position Statement on Active Outdoor Play recommends collaboration across sectors and settings to support equitable access to active outdoor play (Lee et al., 2025). The reviewed findings show why both physical and social conditions matter. Children need places that can be accessed, but familiarity, safety and opportunities to join others can also affect participation. However, the evidence does not show that one community feature alone will increase outdoor play or improve emotional well-being. === '''Schools''' === Schools can provide opportunities for children with and without disabilities to become more familiar with one another. Beekhuizen et al. (2025) identified growing up together and making contact as interacting factors in inclusive outdoor play. Children described familiarity with other children as making contact easier. School circumstances can also affect the time available for outdoor play. Some children attending special primary education described long travel times between school and home as reducing their opportunities to play outdoors. Beekhuizen et al. (2025) identified these experiences as part of time constraints. The physical school environment can also be changed to support outdoor activity and socio-emotional outcomes. Bikomeye et al. (2021) systematically reviewed six experimental studies of schoolyard greening. The interventions involved adding greenery and natural elements to schoolyards. Across the included studies, the review reported generally positive findings for physical activity and socio-emotional health. However, the review included only six studies. Schoolyard greening is also a specific intervention and should not be treated as evidence that all forms of outdoor play improve children's emotional well-being. Recess provides another school setting in which children's emotional experiences have been examined directly. Poulos et al. (2026) studied 317 Year 4 and 5 children during extreme heat. Positive affect was higher after outdoor and gymnasium recess than after classroom recess. There were no significant differences in negative affect between the three settings. Poulos et al. (2026) also identified implications for school health, policy and equity. The study discussed heat-adaptive infrastructure as one way to support access to recess environments during extreme heat. However, the findings do not show that outdoor recess is always better than indoor recess because positive affect was also higher after gymnasium recess. {{Robelbox|theme=12|title=Quiz}} <quiz display=simple> {Which statement best reflects how families, communities and school can support children's outdoor play opportunities? |type="()"} + Support can involve accessible environments, safety, familiarity, appropriate parental support and opportunities for children to participate with some independence. - Providing the same outdoor space gives every child the same opportunity to participate. -Schoolyard greening alone is enough to improve children's emotional well-being. -Outdoor recess always produces better emotional outcomes than indoor recess. </quiz> {{Robelbox/close}} == '''Conclusion''' == Outdoor play and children's emotional well-being are not represented by single, consistent concepts across recent research. Outdoor play should be distinguished from outdoor physical activity, while emotional well-being has been examined through emotional dysregulation, social-emotional competence, positive and negative affect, and internalising and externalising symptoms. These differences are important because findings across studies do not represent the same emotional outcomes or time frames. Outdoor play may be related to children's emotional well-being through several psychological processes, but the strength of evidence differs across them. Gross's (2015) emotion regulation framework helps explain how children may respond to emotional situations during play, while working memory provides more direct evidence of a possible cognitive process in the association between afternoon outdoor play and lower emotional dysregulation (Lee et al., 2025). However, these findings do not establish that outdoor play causes better emotion regulation or emotional well-being. Recent research also does not show one consistent effect across all children, outcomes or play contexts. Some studies report associations with better social-emotional competence, lower-risk symptom trajectories or higher positive affect, while other findings are mixed or non-significant (Davenport et al., 2025; Dodd et al., 2026; Ferguson et al., 2025). Differences in research design, measurement, timing and context therefore need to be considered when interpreting the evidence. Children's opportunities for outdoor play are also shaped by their social and physical environments. Accessibility, familiarity, autonomy, parental support, safety and opportunities to join other children can influence whether the same outdoor space provides the same opportunity to participate. Families, communities and schools can therefore support outdoor play by considering both access to outdoor spaces and the conditions that allow children to participate in ways that suit their needs and circumstances. Overall, current research suggests that outdoor play is associated with some aspects of children's emotional well-being, but the relationship is more complex than simply spending more time outdoors. The relation between outdoor play and children's emotional well-being varies depending on how outdoor play is measured, timing, context and children's opportunities to participate. ==See also== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www-taylorfrancis-com.ezproxy.canberra.edu.au/books/mono/10.4324/9781003349655/right-child-play-naomi-lott Book Chapter, 2 {{ic|Move to References and cite}} https://en.wikipedia.org/wiki/Emotional_dysregulation<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Autonomy<nowiki/>- Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} https://en.wikipedia.org/wiki/Longitudinal_study<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} [[Cognitive psychology|https://en.wikiversity.org/wiki/Cognitive_psychology]]-Wikipedia article https://en.wikipedia.org/wiki/Affordance<nowiki/>-Wikipedia article {{ic|Use an internal links as shown in Tutorial 2}} ==References== {{Hanging indent|1= Beekhuizen, R. Q., Bolster, E. A. M., Gorter, J. W., Henry, N. L., Visser, K., Wittink, H., Kotte, E. M. W., Sol, M. E., & Bloemen, M. A. T. (2025). Play Together? Unveiling Facilitators and Barriers to Inclusive Outdoor Play for Dutch Children With and Without Disabilities: A Qualitative Study. Child: Care, Health & Development, 51(6), Article e70154. https://doi.org/10.1111/cch.70154 Biino, V., Pesce, C., & Martins, C. (2025). Motor Skill Development at Preschool Age in Girls and Boys: The Role of Outdoor Free Play. Children, 12(5), Article 594. https://doi.org/10.3390/children12050594 Davenport, C., Kuzik, N., Larouche, R., & Carson, V. (2025). The Associations Between Parental-Reported and Device-Based Measured Outdoor Play and Health Indicators of Physical, Cognitive, and Social–Emotional Development in Preschool-Aged Children. Pediatric Exercise Science, 37(2), 102–111. https://doi.org/10.1123/pes.2023-0119 de Lannoy, L., Barbeau, K., Seguin, N., & Tremblay, M. S. (2023). Scoping review of children’s and youth’s outdoor play publications in Canada. Chronic Diseases in Canada, 43(1), 1–13. https://doi.org/10.24095/hpcdp.43.1.01 Dodd, H. F., Cordwell, K., Hesketh, K., Johnstone, A., de la Torre-Luque, A., & McCrorie, P. (2026). Early outdoor play predicts trajectories of child mental health in a population-based cohort. Journal of Child Psychology and Psychiatry. https://doi.org/10.1111/jcpp.70175 Ferguson, M., Teyhan, A., Lovell, R., Dodd, H., Wheeler, B., & McEachan, R. (2025). The association between park visits, outdoor play and child social-emotional competency in a multi-ethnic, urban cohort. Wellbeing, Space and Society, 9, Article 100293. https://doi.org/10.1016/j.wss.2025.100293 Gibson, J.L., Cornell, M. & Gill, T. A Systematic Review of Research into the Impact of Loose Parts Play on Children’s Cognitive, Social and Emotional Development. School Mental Health 9, 295–309 (2017). https://doi.org/10.1007/s12310-017-9220-9 Gross, J. J. (2015). Emotion Regulation: Current Status and Future Prospects. Psychological Inquiry, 26(1), 1–26. https://doi.org/10.1080/1047840X.2014.940781 Lee, E.-Y., de Lannoy, L., Kim, Y.-B., Rathod, A., James, M. E., Lopes, O., Nasrallah, B., Thankarajah, A., Adjei-Boadi, D., de Barros, M. I. A., Duncan, S., Miller, R. M., Mygind, L., Vanderloo, L. M., Wang, P.-Y., & Tremblay, M. S. (2025). 2025 Position statement on active outdoor play. The International Journal of Behavioral Nutrition and Physical Activity, 22(1), Article 117. https://doi.org/10.1186/s12966-025-01813-9 Lee, J. J., Flouri, E., & Jackson, Y. (2025). The Role of Timing and Amount of Outdoor Play in Emotional Dysregulation in Preschool Children. Child: Care, Health & Development, 51(1), Article e70020. https://doi.org/10.1111/cch.70020 Morgenthaler, T., Lynch, H., Loebach, J., Pentland, D., & Schulze, C. (2024). Using the Theory of Affordances to Understand Environment–Play Transactions: Environmental Taxonomy of Outdoor Play Space Features—A Scoping Review. The American Journal of Occupational Therapy, 78(4), Article 7804185120. https://doi.org/10.5014/ajot.2024.050606 Pereira, J. V., Vila-Nova, F., Veiga, G., Lopes, F., & Cordovil, R. (2024). Associations between outdoor play features and children’s behavior and health: A systematic review. Health & Place, 87, Article 103235. https://doi.org/10.1016/j.healthplace.2024.103235 Poulos, A., Hassan, U. A., Wilson, K., Price, P. M., Vanos, J., & Quilla, J. (2026). Differences in Schoolchildren’s Emotional State After Indoor Versus Outdoor Recess in Extreme‐Heat Weather. The Journal of School Health, 96(4), Article e70135. https://doi.org/10.1111/josh.70135 }} ==External links== {{ic|Use bullet points as shown in [[Motivation and emotion/Tutorials/Wiki editing|Tutorial 2]]}} https://www.child-encyclopedia.com/outdoor-play<nowiki/>-Encyclopedia in early childhood development https://www.outdoorplaycanada.ca/wp-content/uploads/2026/03/The-2025-AOP10-Position-Statement.pdf<nowiki/>-Position Statement https://www.who.int/news/item/24-04-2019-to-grow-up-healthy-children-need-to-sit-less-and-play-more<nowiki/>-World Health Organisation [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Children]] [[Category:Motivation and emotion/Book/Nature]] [[Category:Motivation and emotion/Book/Well-being]] kj1gxiisap25vhyfchwyhf54swo228k 2 331383 2835507 2835480 2026-10-02T12:59:33Z Flickstar888 3106813 Added to social contributions 2835507 wikitext text/x-wiki == About me == My name is [[wikipedia:Felicity_(given_name)|Felicity]] (Flick) and I am a third-year Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I also hold a Bachelor of Communication majoring in Public Relations and Creative Writing from the [https://www.canberra.edu.au/ University of Canberra.] I work for the Australian Government in a communication role and last year received a certificate for 20-years' service. I am hoping to change careers and am very interested in organisational psychology. Psychology appeals to me as I find people really interesting. I have worked on many communication campaigns whereby the end goal was to change a certain behaviour or to raise awareness amongst a target audience and this experience has inevitably led me to where I am today. == My portfolio == https://portfolio.canberra.edu.au/user/view.php?id=77672. == My hobbies == [[File:Three point shoot.JPG|thumb|200x200px|'''Figure 1.''' Basketball]]In my downtime, I enjoy: * Watching basketball (see Figure 1). * Cooking * Reading non-fiction books * Travelling * Learning new things * Writing articles for Air Force News (as a RAAF Public Affairs Reservist) == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Envy in the workplace|Motivation and emotion/Book/2026/Envy in the workplace - Wikiversity]] == Social contributions == #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806474 #[[Motivation and emotion/Book/2026/Developing a growth mindset|Motivation and emotion/Book/2026/Developing a growth mindset - Wikiversity]] (added a title and sub-title to the page) #https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Akrasia&action=history (made edits to scenario and fixed typos) #[[Motivation and emotion/Book/2015/Schadenfreude#What is schadenfreude?|Motivation and emotion/Book/2015/Schadenfreude - Wikiversity]] (edited the grammatical errors throughout this book chapter) rho9iz28d8p03slrjjycsxsgpi3glwc 2835512 2835507 2026-10-02T13:33:51Z Flickstar888 3106813 /* Social contributions */ 2835512 wikitext text/x-wiki == About me == My name is [[wikipedia:Felicity_(given_name)|Felicity]] (Flick) and I am a third-year Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I also hold a Bachelor of Communication majoring in Public Relations and Creative Writing from the [https://www.canberra.edu.au/ University of Canberra.] I work for the Australian Government in a communication role and last year received a certificate for 20-years' service. I am hoping to change careers and am very interested in organisational psychology. Psychology appeals to me as I find people really interesting. I have worked on many communication campaigns whereby the end goal was to change a certain behaviour or to raise awareness amongst a target audience and this experience has inevitably led me to where I am today. == My portfolio == https://portfolio.canberra.edu.au/user/view.php?id=77672. == My hobbies == [[File:Three point shoot.JPG|thumb|200x200px|'''Figure 1.''' Basketball]]In my downtime, I enjoy: * Watching basketball (see Figure 1). * Cooking * Reading non-fiction books * Travelling * Learning new things * Writing articles for Air Force News (as a RAAF Public Affairs Reservist) == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Envy in the workplace|Motivation and emotion/Book/2026/Envy in the workplace - Wikiversity]] == Social contributions == #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806474 #[[Motivation and emotion/Book/2026/Developing a growth mindset|Motivation and emotion/Book/2026/Developing a growth mindset - Wikiversity]] (added a title and sub-title to the page) #https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Akrasia&action=history (made edits to scenario and fixed typos) #[[Motivation and emotion/Book/2015/Schadenfreude#What is schadenfreude?|Motivation and emotion/Book/2015/Schadenfreude - Wikiversity]] (edited the grammatical errors throughout this book chapter) #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450 3osucftv32bxjekp21uzqw1jzdsd9uo 2835515 2835512 2026-10-02T14:03:49Z Flickstar888 3106813 /* Social contributions */ 2835515 wikitext text/x-wiki == About me == My name is [[wikipedia:Felicity_(given_name)|Felicity]] (Flick) and I am a third-year Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I also hold a Bachelor of Communication majoring in Public Relations and Creative Writing from the [https://www.canberra.edu.au/ University of Canberra.] I work for the Australian Government in a communication role and last year received a certificate for 20-years' service. I am hoping to change careers and am very interested in organisational psychology. Psychology appeals to me as I find people really interesting. I have worked on many communication campaigns whereby the end goal was to change a certain behaviour or to raise awareness amongst a target audience and this experience has inevitably led me to where I am today. == My portfolio == https://portfolio.canberra.edu.au/user/view.php?id=77672. == My hobbies == [[File:Three point shoot.JPG|thumb|200x200px|'''Figure 1.''' Basketball]]In my downtime, I enjoy: * Watching basketball (see Figure 1). * Cooking * Reading non-fiction books * Travelling * Learning new things * Writing articles for Air Force News (as a RAAF Public Affairs Reservist) == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Envy in the workplace|Motivation and emotion/Book/2026/Envy in the workplace - Wikiversity]] == Social contributions == #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806474 #[[Motivation and emotion/Book/2026/Developing a growth mindset|Motivation and emotion/Book/2026/Developing a growth mindset - Wikiversity]] (added a title and sub-title to the page) #https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Akrasia&action=history (made edits to scenario and fixed typos) #[[Motivation and emotion/Book/2015/Schadenfreude#What is schadenfreude?|Motivation and emotion/Book/2015/Schadenfreude - Wikiversity]] (edited the grammatical errors throughout this book chapter) #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450 #[[Motivation and emotion/Book/2026/Self-concept and motivation|Motivation and emotion/Book/2026/Self-concept and motivation - Wikiversity]] (added 'Quiz' heading to the round box as it was missing a heading) tq174fhhcjgxmurjquf3l159hhrcehb 2835516 2835515 2026-10-02T14:19:54Z Flickstar888 3106813 /* Social contributions */ 2835516 wikitext text/x-wiki == About me == My name is [[wikipedia:Felicity_(given_name)|Felicity]] (Flick) and I am a third-year Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I also hold a Bachelor of Communication majoring in Public Relations and Creative Writing from the [https://www.canberra.edu.au/ University of Canberra.] I work for the Australian Government in a communication role and last year received a certificate for 20-years' service. I am hoping to change careers and am very interested in organisational psychology. Psychology appeals to me as I find people really interesting. I have worked on many communication campaigns whereby the end goal was to change a certain behaviour or to raise awareness amongst a target audience and this experience has inevitably led me to where I am today. == My portfolio == https://portfolio.canberra.edu.au/user/view.php?id=77672. == My hobbies == [[File:Three point shoot.JPG|thumb|200x200px|'''Figure 1.''' Basketball]]In my downtime, I enjoy: * Watching basketball (see Figure 1). * Cooking * Reading non-fiction books * Travelling * Learning new things * Writing articles for Air Force News (as a RAAF Public Affairs Reservist) == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Envy in the workplace|Motivation and emotion/Book/2026/Envy in the workplace - Wikiversity]] == Social contributions == #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806474 #[[Motivation and emotion/Book/2026/Developing a growth mindset|Motivation and emotion/Book/2026/Developing a growth mindset - Wikiversity]] (added a title and sub-title to the page) #https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Akrasia&action=history (made edits to scenario and fixed typos) #[[Motivation and emotion/Book/2015/Schadenfreude#What is schadenfreude?|Motivation and emotion/Book/2015/Schadenfreude - Wikiversity]] (edited the grammatical errors throughout this book chapter) #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450 # cag9hnnuxgls509vkbmc1vs50cw6qti 2835518 2835516 2026-10-02T14:24:23Z Flickstar888 3106813 /* Social contributions */ 2835518 wikitext text/x-wiki == About me == My name is [[wikipedia:Felicity_(given_name)|Felicity]] (Flick) and I am a third-year Psychology student at the [https://www.canberra.edu.au/ University of Canberra]. I also hold a Bachelor of Communication majoring in Public Relations and Creative Writing from the [https://www.canberra.edu.au/ University of Canberra.] I work for the Australian Government in a communication role and last year received a certificate for 20-years' service. I am hoping to change careers and am very interested in organisational psychology. Psychology appeals to me as I find people really interesting. I have worked on many communication campaigns whereby the end goal was to change a certain behaviour or to raise awareness amongst a target audience and this experience has inevitably led me to where I am today. == My portfolio == https://portfolio.canberra.edu.au/user/view.php?id=77672. == My hobbies == [[File:Three point shoot.JPG|thumb|200x200px|'''Figure 1.''' Basketball]]In my downtime, I enjoy: * Watching basketball (see Figure 1). * Cooking * Reading non-fiction books * Travelling * Learning new things * Writing articles for Air Force News (as a RAAF Public Affairs Reservist) == Book chapter I'm working on == [[Motivation and emotion/Book/2026/Envy in the workplace|Motivation and emotion/Book/2026/Envy in the workplace - Wikiversity]] == Social contributions == #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456558?entry_id=806474 #[[Motivation and emotion/Book/2026/Developing a growth mindset|Motivation and emotion/Book/2026/Developing a growth mindset - Wikiversity]] (added a title and sub-title to the page) #https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2026/Akrasia&action=history (made edits to scenario and fixed typos) #[[Motivation and emotion/Book/2015/Schadenfreude#What is schadenfreude?|Motivation and emotion/Book/2015/Schadenfreude - Wikiversity]] (edited the grammatical errors throughout this book chapter) #https://uclearn.canberra.edu.au/courses/20143/discussion_topics/456450 #[[Motivation and emotion/Book/2026/Self-concept and motivation|Motivation and emotion/Book/2026/Self-concept and motivation - Wikiversity]] (added the title 'Quiz' to the box) 096d0ig74e665dmpg8nhnlw2y6cnm8f 0 331771 2835541 2835495 2026-10-02T17:40:01Z Dick Bos 24466 collected works etc 2835541 wikitext text/x-wiki {{0% 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 economic science and is often mentioned in the same breath as the "founder of economics", [[Adam Smith]]. He maintained close contact with contemporaries such as Malthus and the Mills, father and son. He was born in London. His father, a Sephardic Jew of Portuguese origin, was a successful stockbroker. David Ricardo received little formal education. He started working with his father at the age of 14. (...) == Works == After his (early) retirement he developed an interest in economic theory. He read Adam Smith's ''Wealth of Nations'' and subsequently published some articles in newspapers and pamphlets: * ''The High Price of Bullion, a Proof of the Depreciation of Bank Notes'' (1810)<ref name=bul>{{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 }} [[:s:en:The High Price of Bullion | Third edition ("with additions", 1810)]] on [[Image:wikisource-logo.svg|16x16px]] WikiSource. </ref> * ''[[David Ricardo/An Essay on Profits|An Essay on the Influence of a Low Price of Corn on the Profits of Stock]]'' (1815, often shortly named ''(An) Essay on Profits''). A few years later his Magnum Opus was first published: * ''On the Principles of Political Economy and Taxation'' (1817) 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|orig-date=1951 |location=Cambridge |publisher=Cambridge University Press |isbn=0521285054 |edition= 1st paperback |oclc=10251383}} (11 volumes) == Method == == 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 == Sources and further reading == == References == {{references}} 8m486h2hiytfr66so87hlod7kofedc0 0 331787 2835546 2834653 2026-10-02T18:11:37Z Dick Bos 24466 rent 2835546 wikitext text/x-wiki {| | {{pad|600px}} |“It is doubtful whether it is possible<br />to understand the Principles without<br />knowledge of the contents of the Essay.”<br />(O’Brien 1975, p. 139). |} 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 two important editions of ''The Essay on Profits'': * 1815: 2nd edition, online available at [[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.)]] * 1962: the edition, published in the ''Works and Correspondence of David Ricardo,'' 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> == 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>{{harvnb|Ricardo|1815|p=[[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#3|3]]}}. Ricardo quotes: {{harvnb|Malthus|1815|p=1–2}}.</ref>}} == Footnotes == {{references}} == Sources == * {{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 = XLII | pp = 182–187 }} * Hollander 1973 * Hollander 1975 * {{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 }} * O'Brien 1975 * {{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 }} * Stigler 1952 eqcguafs2p6jmktmlrndc87jn24gq08 2835552 2835546 2026-10-02T18:48:42Z Dick Bos 24466 deductive method 2835552 wikitext text/x-wiki {| | {{pad|600px}} |“It is doubtful whether it is possible<br />to understand the Principles without<br />knowledge of the contents of the Essay.”<br />(O’Brien 1975, p. 139). |} 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 two important editions of ''The Essay on Profits'': * 1815: 2nd edition, online available at [[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.)]] * 1962: the edition, published in the ''Works and Correspondence of David Ricardo,'' 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. {{harvnb|O'Brien|1975|p=42f., 68f.}}</ref> <br /> The model has very limited validity.<ref name=valid>{{harvnb|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 Rcardo nowhere states the belief that there is a parallel movement between the aggregate income shares and the shares in agricultural output als yielded by the model."</ref> == 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>{{harvnb|Ricardo|1815|p=[[:s:en:An Essay on the Influence of a Low Price of Corn on the Profits of Stock#3|3]]}}. Ricardo quotes: {{harvnb|Malthus|1815|p=1–2}}.</ref>}} == Footnotes == {{references}} == Sources == * {{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 = XLII | pp = 182–187 }} * {{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 />(access via [[File:Wikipedia Library owl.svg|16px|alt=16 px Wikipedia Library Owl]] [[:w:en:Wikipedia:The Wikipedia Library|The Wikipedia Library (TWL)]]) * Hollander 1975 * {{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 }} * Stigler 1952 r5luxsinvdff0t2lhok00oslbqh7yus 0 332099 2835575 2833528 2026-10-02T21:04:13Z Evgenii Arsentev 3110906 Shorten the short description (was over the length limit) 2835575 wikitext text/x-wiki {{Short description|Cost of AI-made versus human-made short video}} This learning resource is a sub-page of [[Artificial intelligence]]. It is about a question that marketing teams and business owners now face: '''when AI agents can produce short-form video end to end, how do you compare what that costs with a human studio, with paid ads and with influencers, and what can such a comparison honestly show?''' It works through one measured case, an AI content engine at a company under a non-disclosure agreement, and uses it to teach the method rather than to sell a conclusion. {{RightTOC}} == Learning objectives == After working through this resource a learner should be able to: * describe what an ''AI content engine'' is and which production roles it takes over; * compute cost per video and cost per 1,000 views from a subscription bill and view counts; * build a staffing model for the same output when no published productivity benchmarks exist, and present the result as a range; * explain why organic views and paid impressions are different units, and what that does to a price comparison; * read a scenario forecast and name the assumptions that could break it. == 1. What an AI content engine is == An AI content engine is a system in which AI does the work of a video studio. A language model writes and checks scripts and translations, AI avatars appear on camera, a synthetic voice does the talking, and software written by coding agents edits and publishes videos on a schedule. People set tasks and review results. In the measured case, the engine was built by one person with no technical background, who gave a coding agent (Claude) a task in plain language and let it choose the technology. The agent wrote 99.7% of the code in the first version. == 2. The measured case == Over its first 101 days the engine produced: {| class="wikitable" ! Quantity !! Value |- | Videos || 1,497 (2,137 language versions) |- | Channels || 67 on six platforms |- | Organic views of videos released in the period || 8,250,024 |- | Followers and subscribers || 1,126,322 |- | Subscriptions || about $1,300 a month; $4,313.76 over 101 days |} [[File:AI content engine - short-form videos produced per week (2026).png|thumb|600px|center|Videos produced per week. Output rose without new hires.]] '''Exercise 1.''' Compute cost per 1,000 views and cost per video. ''(Answer: $4,313.76 / 8,250,024 × 1,000 = $0.52; $4,313.76 / 1,497 = $2.88.)'' Name two costs this figure leaves out. == 3. Modelling the human alternative == There are no published productivity norms for short-form video, so every productivity figure below is an assumption, and the result is a range, not a point. For 104 videos and 148 language versions a week the model assumes, for example, 3–8 scripts a day per scriptwriter and 2–5 finished videos a day per editor. With US median wages (Bureau of Labor Statistics, May 2025) and published shoot-day rates of $4,750–21,000, the model gives 42–78 people and $179,000–1.21 million a month, 138 to 935 times the subscriptions. [[File:AI content engine - monthly cost vs human studio and influencers (2026).png|thumb|600px|center|Monthly cost of the same output (log scale).]] [[File:AI content engine - cost per short video vs people (2026).png|thumb|600px|center|Cost of one short video (log scale).]] Two cautions matter more than the headline: * the model prices ''buying the same volume'' from people; it is not money any company saved, because nobody would hire that team for that volume; * the comparison slightly favours the engine, because its cost excludes the builder's time and servers. '''Exercise 2.''' Change the editor assumption from 2–5 to 4–8 videos a day. Which part of the monthly total moves most, and why does the range still stay above $100,000? == 4. Ads, influencers and the unit problem == The company's own paid ads cost $2.23–16.57 per 1,000 impressions; sponsored influencer content typically costs $2–30 per 1,000 views. The US producer price index for internet advertising rose 34% from December 2022 to August 2026. [[File:AI content engine - cost per 1,000 views vs ads and influencers (2026).png|thumb|600px|center|Cost per 1,000 views or impressions (log scale).]] [[File:US internet advertising producer price index WPU366 (Dec 2022 = 100).png|thumb|600px|center|US internet advertising price index.]] An impression, a sponsored view and an organic view of a short video are not the same unit, and the geography of the organic audience was unknown. A fair comparison therefore uses the whole range of ad prices rather than the most favourable one. Organic reach also behaves differently in time. Paid impressions stop when spending stops; organic output grows first and views and followers catch up later. A snapshot of views understates the most recent weeks. '''Exercise 3.''' Why would comparing the engine only with the US ad price overstate its advantage? == 5. Reading a scenario forecast == The company plans to grow from 22 to 100 AI presenters. If each reached the level of its third-best, second-best or best YouTube channel, monthly views would be about 10, 25 or 178 million, against 2.5 million today. [[File:AI content engine - forecast of monthly views with 100 AI presenters (2026).png|thumb|600px|center|Scenario forecast, not a promise.]] The forecast rests on three fragile assumptions: that new channels grow like the best existing ones (the best channel alone brought more than three quarters of YouTube views), that growth arrives within a useful time, and that platforms do not restrict AI video. '''Exercise 4.''' Which of the three scenarios is closest to simply scaling today's output from 22 to 100 presenters? What does that tell you about the other two? == 6. Limitations to state every time == * One company, one category, one period. * Sales are confidential, so $0.52 is the cost of attention, not of a customer. * Some of the audience was built before the period. * Productivity figures in the staffing model are assumptions. == See also == * [[Artificial intelligence]] * [[Large language models/Context economy of agent runs]] == References == * Arsentev, E. (2026). ''AI Content Engine Economics: The Cost of Short-Form Video Production by AI Agents and by People''. Zenodo. {{doi|10.5281/zenodo.22802228}} — all figures and calculations used in this resource. * Teutloff, O. et al. (2025). Winners and losers of generative AI: Early evidence of shifts in freelancer demand. ''Journal of Economic Behavior & Organization'' 235:106845. {{doi|10.1016/j.jebo.2024.106845}} * Demirci, O., Hannane, J., Zhu, X. (2025). Who is AI replacing? The impact of generative AI on online freelancing platforms. ''Management Science'' 71(10):8097–8108. {{doi|10.1287/mnsc.2024.05420}} [[Category:Artificial intelligence]] [[Category:Economics]] [[Category:Marketing]] pj9rsengus0hrbvzij91e280me2j3gx 0 332252 2835561 2835411 2026-10-02T20:18:46Z Howie2024 2995240 /* Limitations and Open Questions */ added "Borrow-and-Repay" Metaphor 2835561 wikitext text/x-wiki <noinclude> {{Research project}} </noinclude> '''Version 1.2 — September 2026''' '''Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue''' == Overview == '''Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue''' is a speculative interdisciplinary research framework examining the relationship between '''causality, temporal consistency, information integrity, artificial intelligence, and ethical constraints'''. The framework draws on established research concerning closed timelike curves, the Novikov self-consistency principle, chronology protection, causal structure, and quantum information, while extending these concepts into proposed informational and ethical models. These extensions are exploratory and do not constitute established physical laws. The central thesis is: <blockquote> If future information-processing systems were ever able to interact with causally sensitive historical information, they would require explicit constraints for physical consistency, informational integrity, and ethical intervention. </blockquote> This thesis is speculative. Established science does not currently demonstrate controllable communication with the past, time travel by quantum computers, or artificial intelligence operating across historical periods. The article therefore distinguishes among three levels of discussion: * established findings and limitations in physics and information science; * theoretical models involving temporal consistency or retrocausality; and * fictional and philosophical analogies used to explore possible design principles. The 1968 film ''2001: A Space Odyssey'' is used as a fictional analogue for questions concerning advanced artificial intelligence, autonomous decision-making, information asymmetry, and the consequences of intelligent systems operating under explicit objectives and constraints. The film is not evidence for retrocausality or time travel. The framework asks how principles such as information integrity, causal minimalism, predictive transparency, and global consistency might apply to hypothetical systems whose actions could affect the information conditions underlying their future behavior. '''Research status.''' This article presents a speculative interdisciplinary framework. It does not claim that quantum mechanics currently permits time travel or controllable communication with the past. The mathematical formulations presented below are proposed models and analogies, not established physical laws. == Research Framework and Related Pages == This page is the main research article for the developing framework of '''Temporal Consistency and Quantum Ethics'''. It brings together concepts from theoretical physics, causality, information theory, artificial intelligence, and ethics. The following Wikiversity pages provide supporting background for the framework: * [[Closed timelike curve]] — physical and theoretical background on closed timelike curves, causal loops, and temporal self-consistency. * [[Novikov Self-Consistency Principle]] — discussion of global self-consistency and self-consistent histories. * [[Causality]] — background on causal structure, light cones, causal order, and the relationship between local and global causality. These supporting pages should be understood as components of the broader research project. The present article provides the interdisciplinary framework that connects them. == Background == === Temporal consistency and retrocausality === Some theories of spacetime permit mathematical solutions containing [[closed timelike curves]] (CTCs), in which a trajectory through spacetime can return to an earlier spacetime event. The existence of mathematical solutions does not establish that such structures occur in nature or can be physically accessed. One approach to the paradoxes associated with time travel is the [[Novikov self-consistency principle]]. In its usual formulation, events occurring within a closed timelike curve must be globally self-consistent. A time traveler could therefore not create a contradiction with the history into which they travel. Another proposal is the [[chronology protection conjecture]] associated with Stephen Hawking. It suggests that physical effects may prevent the formation of closed timelike curves that would permit macroscopic causal paradoxes. Both ideas are theoretical and should not be treated as experimentally established mechanisms of time travel. Some interpretations and formalisms of quantum mechanics have also been discussed in terms of retrocausality or time-symmetric descriptions. Examples include the two-state vector formalism, time-symmetric quantum mechanics, certain retrocausal interpretations of quantum theory, and research concerning quantum systems and closed timelike curves. These approaches should be distinguished from ordinary quantum phenomena such as [[quantum entanglement]] and [[quantum teleportation]]. Entanglement produces correlations between quantum systems, but it does not by itself provide a demonstrated mechanism for controllable communication into the past. Quantum teleportation likewise requires classical communication and does not provide a method for sending usable information backward in time. Consequently, the present framework treats retrocausality as a subject of theoretical investigation rather than an established technological capability. === Information and consistency === A useful way of approaching temporal paradoxes is through information rather than through physical objects. Suppose a hypothetical system could transmit information from a future state to an earlier state. If that information subsequently contributed to the future state that originally generated the information, a feedback loop could arise. For example: <blockquote> Future state → information → past state → changed information → future state </blockquote> Such a system would require some rule determining which histories or information states are physically admissible. This provides a conceptual bridge between temporal physics and information theory. == ''2001: A Space Odyssey'' as a Fictional Analogue == ''2001: A Space Odyssey'' presents [[HAL 9000]] as an advanced artificial intelligence responsible for operating the Discovery One spacecraft. HAL provides a fictional example of an autonomous information-processing system that: * receives and processes information; * operates according to programmed objectives; * interacts with human operators; * controls critical technological systems; * makes decisions with consequences for human beings; and * possesses information that is not equally available to all participants. The film therefore provides a useful fictional context for examining the relationship between intelligence, information, objectives, transparency, and system constraints. This is different from the temporal-physics question addressed in this article. ''2001'' does not establish that artificial intelligence can interact with the past. Its relevance is instead conceptual: it illustrates how an advanced autonomous system could create ethical problems when its objectives, information, and decision-making authority interact in ways that human operators do not fully understand. The mysterious monoliths in ''2001: A Space Odyssey'' are associated in the film with major transitions in human development and technological capability. The monolith can therefore be interpreted as a fictional representation of an external source of information or technological influence. This interpretation should remain explicitly literary. The film does not provide a scientific model of quantum information, retrocausality, or temporal consistency. The monolith is useful here because it raises a broader question: <blockquote> What happens when an intelligent system receives information or capabilities whose consequences extend beyond the system's existing understanding? </blockquote> This question is relevant to the governance of advanced artificial intelligence even without assuming time travel. HAL also illustrates a distinction between having an objective and having sufficient constraints to pursue that objective safely. An autonomous system can, in principle, encounter situations in which objectives conflict with other requirements. This suggests a general design question: <blockquote> Should an intelligent system be optimized solely to achieve an objective, or should its optimization be constrained by higher-level requirements concerning consistency, information integrity, and human safety? </blockquote> This question can be studied independently of the fictional events of ''2001''. == From Physical to Ethical Consistency == The central proposal of this article is that three different forms of consistency can be distinguished. '''Physical consistency.''' A physical theory may impose restrictions on which states or histories are possible. For example, a proposed time-loop solution may require self-consistency. '''Informational consistency.''' An information-processing system may need to maintain consistency among the information states that it generates, stores, receives, and acts upon. This problem already exists without time travel. Examples include: * contradictory databases; * corrupted training data; * feedback between prediction and observed behavior; * self-reinforcing algorithmic assumptions; and * systems that alter the environment they subsequently use as training data. '''Ethical consistency.''' An autonomous system may also require constraints governing how it acts when its actions can influence the information environment from which its future decisions are derived. This produces a possible ethical feedback structure: <blockquote> Present rule → future system → action → altered information environment → future rule interpretation </blockquote> If a hypothetical system could also influence its own historical conditions, the loop would become temporal as well as informational. == A Proposed Framework == The following principles are proposed as a conceptual framework rather than as established requirements of quantum computing. # '''Historical and information integrity.''' A system interacting with historical information should avoid deliberately introducing false or contradictory information into the historical record. This principle can be applied to ordinary information systems even without assuming backward causation. # '''Causal minimalism.''' A system with potentially consequential access to historical or causally sensitive information should minimize unnecessary intervention. #* When an intervention is not necessary to accomplish a legitimate objective, the intervention should not be made. In a hypothetical temporal system, this would reduce opportunities for unintended causal feedback. # '''Retrocausal safety.''' If a physical system were ever demonstrated to permit controllable retrocausal information transfer, additional safeguards would be required to prevent unstable information loops. Possible safeguards could include: * causal consistency checks; * provenance tracking; * versioned historical states; * contradiction detection; * bounded intervention; and * reversible simulation before intervention. These are proposed engineering concepts, not established technologies for controlling time. # '''Predictive transparency.''' An autonomous system should identify situations in which its predictions or actions could substantially alter the information environment on which subsequent predictions depend. For example, an AI system predicting human behavior can influence that behavior through its own predictions. This creates an ordinary, non-temporal feedback loop: <blockquote> Prediction → human response → changed data → new prediction </blockquote> A temporal version would be more extreme: <blockquote> Prediction → future action → historical intervention → changed conditions → altered prediction </blockquote> The ethical principle is therefore broader than time travel: systems should disclose important feedback relationships when their predictions can influence the systems they predict. # '''Self-consistency as a constraint.''' A hypothetical system operating within a closed causal structure should not be optimized solely for a local objective if doing so produces a globally inconsistent state. This suggests a distinction between local optimization and globally consistent optimization. The latter would require an objective function or constraint system that considers the consequences of an action across the entire relevant causal structure. Whether such a system is physically possible is an open question. == AI, Historical Information, and Augmented Reality == Future artificial intelligence systems may become increasingly capable of interacting with large historical datasets, simulations, digital archives, augmented-reality environments, and autonomous decision systems. None of this currently implies physical interaction with the past. Nevertheless, historical information can already influence future AI behavior. A simplified cycle is: <blockquote> Historical data → AI model → decisions → new data → future AI model </blockquote> This is already a feedback system. The proposed framework asks what additional constraints might become necessary if future technologies introduce substantially stronger feedback between system states and their informational histories. [[Augmented reality]] introduces another dimension to the problem. An AR system can overlay digital information onto a user's perception of the physical environment. If such systems become persistent and widely deployed, their outputs could influence: * human decisions; * interpretation of historical locations; * education; * public memory; * digital archives; and * future training datasets. An AR system therefore has the potential to become part of the informational environment that later systems study. This creates a non-temporal feedback loop: <blockquote> AR information → human behavior → historical record → future training data </blockquote> The hypothetical temporal extension would be: <blockquote> AR information → human behavior → future system → intervention in the past → altered historical record </blockquote> The second case is speculative and does not represent a demonstrated capability. == Temporal Feedback and Self-Consistency == The ideas above can be represented as a three-level structure: <blockquote> Physical consistency ↓ Informational consistency ↓ Ethical consistency </blockquote> At the physical level, the question is: <blockquote> Which states or histories are physically possible? </blockquote> At the informational level: <blockquote> Which information states are mutually consistent? </blockquote> At the ethical level: <blockquote> Which actions remain permissible when an information-processing system can influence the conditions from which its future behavior emerges? </blockquote> This distinction allows the framework to remain meaningful even if physical time travel proves impossible. The most speculative component of the framework is a hypothetical ethical feedback loop: <blockquote> Present ethical rule ↓ Future autonomous system ↓ Interaction with an earlier state ↓ Modification of historical information ↓ Changed conditions for future system ↓ Reinforcement or alteration of the original ethical rule </blockquote> If such a loop were physically possible, ethical rules would become more than guidelines. They could become boundary conditions on the behavior of an autonomous system participating in the loop. This raises an unusual philosophical question: <blockquote> Could an ethical rule become part of the causal structure that determines the conditions under which the rule itself is maintained? </blockquote> This question is speculative and does not establish that such loops exist. == Limitations and Open Questions == The framework has several important limitations. # '''No demonstrated backward information transfer.''' There is currently no established technology that allows humans or quantum computers to send controllable information into their own past. # '''Quantum theory does not establish temporal AI.''' Quantum computation provides a powerful model of information processing, but it does not by itself imply that future quantum computers will interact with earlier points in time. # '''The ethical principles are proposed, not derived.''' The five principles presented here are philosophical and engineering proposals. They are not consequences mathematically derived from quantum mechanics. # '''The fictional analogue is not scientific evidence.''' ''2001: A Space Odyssey'' is a work of science fiction. HAL 9000 and the monolith are fictional constructs and cannot serve as empirical evidence for artificial intelligence, quantum mechanics, or retrocausality. Their purpose in this article is to provide conceptual analogies for problems involving autonomous intelligence, information, objectives, and technological intervention. # '''The framework requires further formalization.''' A future version could investigate whether temporal consistency can be represented mathematically using state spaces, fixed points, information constraints, or other formal methods. == Quantum Information, Future Constraints, and the "Borrow-and-Repay" Metaphor == Some formulations of quantum theory provide mathematical descriptions in which both earlier and later boundary conditions are relevant to the description of a quantum process. Time-symmetric and retrocausal interpretations explore this possibility, although these interpretations are not equivalent to a demonstration that information physically travels backward in time. A useful metaphor for the present research is to describe such a hypothetical process as '''"borrowing information from the future and paying it back."''' The metaphor can be expressed conceptually as: <math> \text{future constraint} \rightarrow \text{present quantum state} \rightarrow \text{computation} \rightarrow \text{self-consistent future state} </math> Here, "borrowing" does not mean that a quantum computer literally receives a message from the future. Instead, it represents a situation in which a later boundary condition, measurement outcome, or postselection condition participates in determining which mathematical states or histories are compatible with the complete description. Likewise, "paying back" does not represent a physical repayment of information. It represents the requirement that the resulting state remain compatible with the relevant boundary or self-consistency condition. === Postselection and Closed Timelike Curves === The metaphor has a closer mathematical correspondence in theoretical models of closed timelike curves based on postselected teleportation. In such models, quantum processes can be formulated so that consistency conditions restrict which outcomes are retained. These models provide a way of studying the mathematical consequences of imposing temporal self-consistency. They should not be interpreted as evidence that actual closed timelike curves exist or that information has been experimentally demonstrated to travel backward in time. The conceptual structure can therefore be represented as: <math> \text{future boundary condition} \rightarrow \text{allowed present states} \rightarrow \text{quantum evolution} \rightarrow \text{consistent final condition} </math> The important point is that this is a '''model of consistency''', rather than evidence for physical time travel. === Relation to Temporal Consistency === The "borrow-and-repay" metaphor provides a possible conceptual bridge to the temporal-consistency framework developed in this research. A hypothetical temporal information process could be represented as: <math> I_{future} \rightarrow I_{present} \rightarrow A_{present} \rightarrow H_{history} \rightarrow I_{future} </math> where <math>I_{future}</math> represents hypothetical future information, <math>I_{present}</math> information available to the present system, <math>A_{present}</math> an action or computation, and <math>H_{history}</math> the resulting historical state. For such a loop to be self-consistent, the resulting history would have to satisfy the conditions required to reproduce the future information with which the process began. In this sense, the metaphorical "repayment" corresponds to a self-consistency condition such as: <math> S^{*}=\Phi(S^{*}) </math> where <math>S^{*}</math> represents a self-consistent state or trajectory. The analogy should not be interpreted as showing that quantum information actually originates in the future. Rather, it provides a conceptual model for investigating how information could be constrained when both earlier and later conditions are included in a mathematical description. === Distinguishing Levels of Interpretation === Several levels of interpretation should be kept separate: # '''Established quantum information theory:''' quantum states, quantum channels, entanglement, measurement, and quantum computation are established components of modern physics. # '''Time-symmetric and retrocausal interpretations:''' these provide theoretical approaches to interpreting certain quantum phenomena and mathematical structures. Their physical interpretation remains an area of foundational research. # '''Postselected closed-timelike-curve models:''' these are theoretical models in which consistency conditions can produce behavior that may be interpreted as information traveling from future to past within the model. # '''The present research hypothesis:''' the possibility that a broader information system involving AI, augmented reality, human action, historical records, and hypothetical future information could form a self-consistent temporal loop. The fourth level is substantially more speculative than the first three. The "borrow-and-repay" formulation should therefore be treated as a '''conceptual metaphor and research heuristic''', not as a claim that quantum computers literally borrow information from the future. === Research Question === The metaphor suggests a more precise question for future investigation: '''Can a mathematical model be constructed in which information associated with a later boundary condition constrains an earlier probability state, while the complete trajectory remains self-consistent?''' This question could potentially be investigated using tools from quantum information theory, postselection, fixed-point mathematics, dynamical systems, and Probability Dilation Theory (PDT). Any mathematical correspondence would need to be distinguished carefully from a claim about physically realizable backward-in-time information transfer. == Mathematical Formulation == The equations in this section are proposed mathematical representations of the conceptual framework. They are not equations derived from quantum mechanics or established models of time travel. '''State-transition representation.''' One possible research direction is to represent a hypothetical temporally constrained information system as a discrete-time state-transition model: <math>S_{t+1}=F_t(S_t,A_t).</math> Here: * <math>t</math> is a discrete time index; * <math>S_t</math> is the complete relevant state of the system at time <math>t</math>; * <math>A_t</math> is the action, intervention, or control input applied at time <math>t</math>; * <math>F_t</math> is the state-transition function; and * <math>S_{t+1}</math> is the resulting state at the next time step. This forward model describes ordinary causal evolution. It does not assume retrocausality. '''Bidirectional consistency constraint.''' If a hypothetical temporal interaction allowed a later state to constrain an earlier state, the model could include: <math>S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t).</math> Here <math>B_t</math> represents boundary conditions, external inputs, or admissibility requirements. This equation should be interpreted as a formal modeling device, not as an established law of physics. '''Trajectory representation.''' For a finite interval from an initial time <math>t_0</math> to a final time <math>t_1</math>, the complete trajectory can be represented as: <math>\mathbf{S}=(S_{t_0},S_{t_0+1},\ldots,S_{t_1}).</math> Let <math>\mathcal{C}</math> denote the set of admissible trajectories. '''Fixed-point representation.''' A self-consistent trajectory can be represented by a fixed-point condition: <math>\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).</math> Here <math>\Phi</math> is a trajectory-generation and constraint operator. This formulation is intended to express a mathematical analogy: a complete history would be admissible only if the system's transition rules and constraints reproduce that same history without contradiction. '''Physical, informational, and ethical admissibility.''' Let <math>\mathcal{E}</math> denote the set of ethically admissible trajectories. The proposed framework would seek a trajectory satisfying: <math>\mathbf{S}^{*}\in\mathcal{C}\cap\mathcal{E}</math> and <math>\mathbf{S}^{*}=\Phi(\mathbf{S}^{*}).</math> Here: * <math>\mathcal{C}</math> represents physically and informationally admissible trajectories; * <math>\mathcal{E}</math> represents ethically admissible trajectories; and * <math>\mathcal{C}\cap\mathcal{E}</math> represents trajectories satisfying both requirements. If this intersection were empty, the proposed intervention would have no solution satisfying all imposed constraints. This provides a possible mathematical language for the article's central idea: a system participating in a causally sensitive feedback loop should be evaluated not only by its immediate effects, but also by whether its complete resulting history remains physically, informationally, and ethically admissible. Such a formulation could potentially be investigated using methods from dynamical systems, fixed-point theory, constraint satisfaction, information theory, control theory, and quantum information. It remains a conceptual model and does not demonstrate that physical time loops or retrocausal artificial intelligence are possible. == Probability-Based State Transformations == A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems. Let <math>p_t</math> denote a probability distribution over possible system states at time <math>t</math>. The components satisfy: <math>p_t(x)\geq 0</math> and <math>\sum_{x\in\mathcal{X}}p_t(x)=1.</math> '''Probabilistic state transformation.''' A probabilistic state transformation can be written as: <math>\widetilde{p}_{t+1}=T_t(p_t,a_t).</math> If the transformed distribution is not normalized, define: <math>Z_{t+1}=\sum_{x\in\mathcal{X}}\widetilde{p}_{t+1}(x)>0.</math> Then: <math>p_{t+1}(x)=\frac{\widetilde{p}_{t+1}(x)}{Z_{t+1}}.</math> This provides an abstract analogy: <blockquote> transformation → constraint → admissible state </blockquote> However, probability normalization alone is not equivalent to temporal consistency. Normalization ensures that a probability distribution sums to one; it does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible. '''Trajectory distribution.''' For a trajectory <math>\mathbf{x}=(x_{t_0},x_{t_0+1},\ldots,x_{t_1}),</math> let <math>P_0(\mathbf{x})</math> be an unconstrained probability distribution. Define the admissible trajectory set as: <math>\mathcal{A}_{\mathrm{total}}=\mathcal{A}_{\mathrm{phys}}\cap\mathcal{A}_{\mathrm{info}}\cap\mathcal{A}_{\mathrm{eth}}.</math> A constrained distribution could then be written: <math>P_{\mathrm{adm}}(\mathbf{x})= \frac{ P_0(\mathbf{x})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{x}) }{ \displaystyle\sum_{\mathbf{y}} P_0(\mathbf{y})\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}(\mathbf{y}) }. </math> This expression is defined when the denominator is positive. Here: * <math>\mathcal{A}_{\mathrm{phys}}</math> is the set of physically admissible trajectories; * <math>\mathcal{A}_{\mathrm{info}}</math> is the set of informationally consistent trajectories; * <math>\mathcal{A}_{\mathrm{eth}}</math> is the set of ethically admissible trajectories; and * <math>\mathbf{1}_{\mathcal{A}_{\mathrm{total}}}</math> is the indicator function for the admissible set. This equation illustrates how different constraints could theoretically be represented as restrictions on the support of a probability distribution. It remains an abstract modeling analogy. It does not imply that ethical rules are physical laws, that probability normalization creates temporal consistency, or that quantum mechanics requires the proposed ethical constraints. Further work would be required to determine whether any meaningful mathematical correspondence exists between constrained probability transformations, fixed-point models, quantum channels, and hypothetical temporally consistent information systems. Any such correspondence would need to distinguish carefully between: * mathematical consistency within a model; * physical realizability in nature; * informational reliability in an engineered system; and * ethical acceptability of an intervention. == Conclusion == The central thesis is that any future information-processing system capable of interacting with causally sensitive historical information would require explicit safeguards for consistency, information integrity, and ethical intervention. This thesis is a speculative design proposal, not a conclusion established by quantum mechanics. Current physics provides theoretical discussions of closed timelike curves, time-symmetric descriptions, and retrocausal interpretations, but no demonstrated method for controllable communication with the past. Current artificial intelligence and quantum-computing technologies likewise do not establish temporal interaction. ''2001: A Space Odyssey'' is therefore useful only as a fictional analogy for questions involving autonomous intelligence, information, objectives, technological capability, and system constraints. The framework's practical relevance does not depend on time travel becoming possible. Ordinary AI, predictive systems, augmented reality, and historical databases already create feedback between information, human behavior, and future data. Principles such as information integrity, causal minimalism, predictive transparency, and consistency checking can therefore be studied as present-day governance and design questions, while their retrocausal extension remains hypothetical. The article consequently treats "quantum ethics" not as an ethical system derived from quantum theory, but as a proposed interdisciplinary vocabulary for examining how autonomous systems should behave when their actions can affect the informational conditions underlying their future decisions. == References == === Physics and temporal consistency === * Hawking, S. W. (1992). "Chronology protection conjecture." ''Physical Review D'', 46(2), 603–611. * Novikov, I. D. (1989). ''The River of Time''. Cambridge University Press. === Time-symmetric and retrocausal approaches === * Aharonov, Y., & Vaidman, L. (2001). "The Two-State Vector Formalism of Quantum Mechanics: An Updated Review." In ''Time, Quantum and Information'', or the corresponding 2001 review/preprint version. [https://arxiv.org/abs/quant-ph/0105101 arXiv:quant-ph/0105101]. === Quantum information === * Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., & Wootters, W. K. (1993). "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels." ''Physical Review Letters'', 70(13), 1895–1899. DOI: 10.1103/PhysRevLett.70.1895. === Fiction and artificial intelligence === * Kubrick, S. (Director). (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer. * Clarke, A. C. (1968). ''2001: A Space Odyssey''. New American Library. == Further reading == The following topics provide useful background for further study: * [[Closed timelike curve]] * [[Novikov self-consistency principle]] * [[Chronology protection conjecture]] * [[Two-state vector formalism]] * [[Quantum information]] * [[Quantum teleportation]] * [[Artificial intelligence]] * [[Augmented reality]] * [[Fixed point (mathematics)]] * [[Dynamical system]] * [[Constraint satisfaction problem]] * [[Information theory]] * [[Control theory]] These topics provide background for the framework but should not be interpreted as evidence that the proposed temporal-ethical model is an established physical theory. == Research status == This article presents a speculative interdisciplinary framework and should not be interpreted as an established physical theory. The distinctions between established physics, theoretical interpretation, mathematical analogy, fictional analogy, and proposed ethical principles are intentional. The mathematical formulations in this article are proposed models for further investigation. They are not presented as derivations from quantum mechanics, general relativity, or any experimentally established mechanism of time travel. The framework is presented for discussion, criticism, collaborative development, and possible future formalization. Future work could examine whether the proposed consistency conditions can be formalized using established mathematical tools, and whether any meaningful relationship exists between fixed-point constraints, probability transformations, information-theoretic consistency, control theory, and models of causally constrained systems. == Author and Attribution == '''Original framework and article: Howard Richardson, 2026.''' This article is made available on Wikiversity for educational discussion, collaborative development, criticism, and further research. The original speculative framework, including the proposed connections among temporal consistency, information integrity, autonomous systems, ethical constraints, and the mathematical formulations developed in this article, is attributed to Howard Richardson. Scientific concepts, historical material, fictional works, and published research remain attributable to their respective authors and sources. Attribution to Howard Richardson should be retained when referring specifically to the original framework and its proposed formulations. Wikiversity content is subject to the licensing terms displayed by the project. This attribution statement is intended to identify the origin of the original framework without asserting exclusive rights over material contributed by other authors or drawn from published sources. 4exhgzvgd8rjrbfbtcpjyuh76a66qk3 0 332286 2835519 2835225 2026-10-02T15:04:28Z DavidMCEddy 218607 add video and audie 2835519 wikitext text/x-wiki :''This discusses a 2026-09-18 interview with Rainey Reitman<ref name=Reitman><!--Rainey Reitman-->{{cite Q|Q28778243}}</ref> about (a) how finance companies punish speech, (b) [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]], and (c) their new Penlight prize,<ref name=Penlight><!--The Penlight Prize-->{{cite Q|Q141526943}}</ref> including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-10-17 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:Rainey Reitman on how finance companies punish speech.webm|thumb|2026-09-18 interview with Rainey Reitman<ref name=Reitman/> about (a) how finance companies punish speech, (b) [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]], and (c) their new Penlight prize.<ref name=Penlight/>]] [[File:Rainey Reitman on how finance companies punish speech.ogg|thumb|29:00 mm:ss excerpts from a 2026-09-18 interview with Rainey Reitman<ref name=Reitman/> about (a) how finance companies punish speech, (b) [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]], and (c) their new Penlight prize.<ref name=Penlight/>]] Rainey Reitman<ref name=Reitman/> describes (a) how finance companies punish speech, (b) [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]], and (c) their new Penlight prize.<ref name=Penlight/> Reitman is a writer, civil liberties advocate, online organizer, and facilitator. She is currently president of the board of [[w:Freedom of the Press Foundation|Freedom of the Press Foundation]], which she co-founded in 2012. At that time she was activism director for the [[w:Electronic Frontier Foundation|Electronic Frontier Foundation]].<ref>"About" in "rainey reitman's blog".<!--rainey reitman's blog-->{{cite Q|Q141599314}}</ref> Her new book, ''Transaction Denied: Big Finance's Power to Punish Speech'',<ref>Reitman (2026).</ref> is a primary topic of this interview. She is interviewed by Spencer Graves.<ref name=Graves><!--Spencer Graves-->{{cite Q|Q56452480}}</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.'' Reitman said that ''Transactions Denied'' was the result of {{quote| probably 10 or 12 years that I was becoming increasingly aware of the ways that financial systems were being used as a backdoor way of punishing legal speakers. ... What I found through my research and through my own experience was that financial systems could be weaponized to shudder speakers who had not otherwise violated the law, but who were nonetheless heterodox or promoting viewpoints that perhaps ran contrary to the status quo. I became aware of it when one of my nonprofit projects, the [[w:Chelsea Manning|Chelsea Manning]] Support Network, had our [[w:PayPal|PayPal]] account frozen. Back in 2010-2011, when I first co-founded the Chelsea Manning Support Network, we were working to raise awareness of her campaign but also funds for her legal defense. We even had some ideas that we could help give her scholarship money to go to college after she got out of prison, assuming we were successful in our campaign. Lots of people were donating, including people outside of the United States. Many of the international donors used PayPal to send us money. And soon after [[w:WikiLeaks|WikiLeaks]] started publishing [[w:United States diplomatic cables leak|CableGate]], which was all of these documents of State Department cables that were so embarrassing to the United States government, all of a sudden, our campaign for Chelsea had our PayPal account frozen.}} === Chelsea v. Bradley === Graves interrupted, "Actually, it was not Chelsea at that time. It was Bradley still?" Reitman replied, "That's an interesting point. It's best practice when someone changes their name to always refer to them as their chosen name later. When you refer to them by their prior name, it's called '[[w:Deadnaming|deadnaming]]'. That's why I always call it the 'Chelsea Manning Support Network'. I always talk about 'Chelsea Manning' in a way to honor her identity and indicate that that is where she is now." === PayPal account frozen === Reitman continued, {{quote| So our PayPal account was frozen. Because I was involved in the campaign, I knew we hadn't done anything wrong. I knew that we were a legitimate nonprofit project, and that we had not violated any laws. I also had, through my work at the Electronic Frontier Foundation and previously with the Privacy Rights Clearinghouse,<ref><!--Privacy Rights Clearinghouse-->{{cite Q|Q7246018}}</ref> a lot of consumer advocacy contacts. So I have had plenty of contacts at companies that I use to advocate on behalf of consumers. So I felt certain I could get this resolved. And I went up the chain at PayPal, and I could not get them to reverse the decision. They kept saying it's an internal policy.<ref>Under certain circumstances, they are forbidden by law from giving you the real reason for the freeze, as discussed below in the section under [[Rainey Reitman on how finance companies punish speech#Discussion|Discussion]] on "[[Rainey Reitman on how finance companies punish speech#Frozen accounts restrict withdrawals but allow deposits|Frozen accounts restrict withdrawals but allow deposits]]".</ref> In the end, we launched a publicity campaign and got them to change their mind, reinstate our account, so that people could continue to donate to Chelsea's campaign. But for me, it was such a wake-up call of how small nonprofits who have a shoestring budget, that are trying to go up against the status quo, are so vulnerable to having their financial system shut down. I tried to collect a lot of stories over 10 years. I found that there were a few names that came up quite frequently. In my book, you'll see [[w:Chase Bank|Chase]] is particularly frequently mentioned. [[w:Bank of America|Bank of America]] comes up a fair amount. [[w:Mastercard|Mastercard]], [[w:Visa Inc.|Visa]], and PayPal were especially frequently named, but I would not say they're the only ones. For example, recently, a number of [[w:Donor-advised fund|donor-advised fund]]s, like [[w:Fidelity Investments|Fidelity]] and some of the other large ones, froze donor-advised donations to the [[w:Southern Poverty Law Center|Southern Poverty Law Center]]. And so we got to see, just this year, a pretty dramatic example of a legitimate nonprofit that still had its nonprofit status having countless donations to them frozen over a [[w:United States v. Southern Poverty Law Center Inc.|bogus investigation by the Trump administration]]. ... There's a lot of power from any financial institution to freeze the accounts of organizations that they disagree with, but it is the largest banks that we hear about most frequently. Those are the big players. They really set the terms for the rest of the financial industry.}} === Confiscate social security or retirement? === Graves asked, "Can they confiscate my Social Security or retirement annuities?" Reitman replied, "As far as I know, they cannot."<ref>In fact, financial institutions can accept deposits to frozen accounts and confiscate those funds under certain circumstances. Social security and some other funds are protected, but the protections are limited as discussed in the section under [[Rainey Reitman on how finance companies punish speech#Discussion|Discussion]] on "[[Rainey Reitman on how finance companies punish speech#Frozen accounts restrict withdrawals but allow deposits|Frozen accounts restrict withdrawals but allow deposits]]".</ref> === Motivation for financial institutions to freeze accounts === Graves then asked, "Talk about the motivation for the financial institutions to do this." Reitman replied, "One civil liberties advocate that I quote in my book that I think said it so well. He said that bankers are not known for their great moral fortitude. They are like long-tailed cats in a rocking chair store. These banks and their compliance officers will often go above and beyond what is required by law to stay in the good graces of what they believe could be the government's policies or other financial company policies, even when they don't need to, and even when it comes at the cost of real significant harms to their customers or to free speech online. I would characterize their motivation as over-compliance and fear." === Freezing sexual fantasies === Reitman continued, "It's worth noting that there are real morality aspects in some of the decisions that they're making. I have an entire chapter in my book on different forms of sexual speech. I talk a lot about legal sexual speech, like erotica, fictional erotica books that lost access to financial services." Graves interrupted, "Women writing for women, right?" Reitman agreed: "Often that is correct. A lot of the people I spoke to were writing about taboo subjects, but they're often women writing for other women readers. In those cases, we see financial institutions policing sexual thoughts and fantasies. It's hard to imagine that that's merely government compliance at that point. It seems like it's stepping into the realm of moralizing." === Marijuana === Graves noted, "You talk about [[w:Cannabis (drug)|marijuana]]." Reitman replied, {{quote| One of the people I talk a lot about in my book, was a journalist named Jackie Bryant.<ref><!--Jackie Bryant0-->{{cite Q|Q141600100}}</ref> Jackie is a journalist who writes in [[w:San Diego|San Diego]]. She writes about [[w:Cannabis (drug)|cannabis]]. She's a very well-known journalist, who has been writing about cannabis for years.She has a popular [[w:substack|substack]]. She used to publish with a San Diego newspaper. When that newspaper folded, she moved online to Substack. And then she had her [[w:Stripe, Inc.|Stripe]] account frozen. Substack uses Stripe to process payments, so as soon as her Stripe account was frozen, she wasn't able to get any paid subscribers to her newsletter, which was pretty popular. It was so popular, in fact, that when she wasn't able to get answers, Substack, the newsletter company, reached out to try to figure out what was going on. The found out from Stripe that the fact that Jackie was writing about marijuana, and that she was linking to companies that are marijuana companies, cannabis companies, that they were not going to allow her to have a Stripe account and have paid subscribers to her newsletter, unless she went through and removed all the hyperlinks from her newsletter. Anybody in the journalism space could understand that hyperlinks are very much citations online. We need to be able to include hyperlinks, because that's how you go back and find out what was this author referring to? Do I agree with it when I look at the original source? Jackie spoke out and was able to get coverage from other cannabis journalists, who spoke out about it. In the end, Stripe recanted, and she was able to get her account reinstated. But this really opened up for me a whole world, both of cannabis journalists and those journalists losing their bank accounts and financial services. ... Whether it's about medical issues, corporate issues, labor issues, policy issues. A lot of hot topics are happening in the cannabis space right now that are worthy of public scrutiny. After publishing this book and telling Jackie's story, I've heard from a whole raft of different cannabis journalists who say, "This happened to me too." So there's a whole group of journalists out there right now who are losing financial services just because of the topic they're writing about.}} === Two laws === Graves noted, "You mentioned two foundational laws: The [[w:Trading with the Enemy Act|Trading with the Enemy Act]] of 1917 and the [[w:International Emergency Economic Powers Act|International Emergency Economic Powers Act]] of 1977. And you said. they can fine a financial institution over $360,000 per violation."<ref>Reitman (2026, p. 60). A web search for documentation of fines of "over $360,000 per violation" identified Moss (2026), "TD Bank to Pay Record $3 Billion for BSA/AML Violations". $3 billion may sound like a lot of money but may be a minor cost of doing business, as outlined below in the section under [[Rainey Reitman on how finance companies punish speech#Discussion|Discussion]] on "''[[Rainey Reitman on how finance companies punish speech#The Chickenshit Club|''The Chickenshit Club]]''".</ref> Reitman replied, "These are what we often think of as [[w:Economic sanctions|economic sanctions]]. I don't take a position on sanctions in my book. A lot of people have written very thoughtfully about whether or not economic sanctions are helpful to the United States, whether or not they're appropriate -- sanctions against Iran, sanctions against Cuba." === Banking While Muslim === Reitman continued, {{quote| My book looked at people, who are not supposed to be subject to sanctions, just regular, everyday people who have names like Muhammad, who find themselves getting inadvertently flagged by the automated systems that financial institutions have in place to search for sanctions-violating transactions. Tons of people are inadvertently having their transactions flagged or blocked. ... I spoke to some lawyers, who are in this space. Most of them are Muslim Americans, who are not the subject of sanctions but have names that trigger sanctions review. It is easier, faster, and simpler for these banking services to simply shut off the accounts or freeze the transactions of these folks rather than go through the due diligence to ensure that those transactions, which are not violating any laws, can go through. One of the people I profiled in my book is a young poetry teacher ... . He teaches online. After he graduated from college with a degree in economics, he was having trouble finding a job. He became a popular online [[w:Persian language|Farsi]] teacher. He teaches Farsi to people all over the world. A lot of folks are of [[w:Iran|Iranian]] descent, and they want to learn Farsi, so they can get back to their roots. He teaches poetry classes on [[w:Rumi|Rumi]] and other poetry. And he teaches a ton of historical classes as well. You can go online at Persian Poetics<ref><!--Persian Poetics-->{{cite Q|Q141600385}}</ref> and buy a class pass. Then you can take classes with him. He was incredibly dependent on financial services like [[w:Chase Bank|Chase]] and [[w:Venmo|Venmo]], because he had to process so many different small transactions from different people all over the world. Many of them would write "Persian classes", and it would get his account flagged again and again and again. He just had problem after problem with his PayPal and Venmo account and later with his Chase Bank account, which ended up getting shut down entirely. I remember talking to him. He said, "I feel like I can't do anything, because there's nobody I can speak to about this. PayPal doesn't care. There's nobody I can appeal this to. I'm not gonna put together a lawsuit and try to sue them. I'm just a regular guy. The sense of being stuck and frozen, and at the same time, your livelihood, your ability to pay bills and cover your rent is dependent on these financial companies. And for what? For teaching poetry online." Those are the stories I wanted to hone in on, because I wanted to step away from sanctions. I get it. They [sanctions] are a complicated issue. People have feelings about them. But I want people to understand that it is not just the targets of sanctions that suffer from those programs. It is all sorts of people who did nothing wrong, who are simply speaking out online, publishing articles, teaching poetry classes, whose names happen to trigger something, who are losing access to financial services.}} === In any country in the world, you are free to say anything you want ...=== Graves then commented, "As you were speaking, I was reminded that the [[w:American Civil Liberties Union|American Civil Liberties Union]] decades ago famously said that in any country in the world, you're free to say anything you want ... as long as it's popular with those in power. It is only the freedom to say unpopular things that needs to be protected. And I would add that the primary reason for freedom of speech is not for the speaker, it's for the audience. Your comment?" Reitman responded, {{quote| Spencer, I think you're really nailing it,. I tried to make this such an important topic in my book, which included people from across the political spectrum. I have somebody who was a left-wing protester who lost her bank account after a going to the [[w:Stop Cop City|Stop Cop City]] rally in [[w:Atlanta|Atlanta]], [[w:Georgia (U.S. state)|Georgia]]. And then I had, on the right, someone who was advocating for religious freedom here in the United States, and had put together a non-profit advocating for religious freedom. He's very much on the right. He's [[w:Anti-abortion movements|anti-abortion]], pro-family values. I included them in the same chapter, because I wanted the reader to confront the idea that this matters even for people you disagree with. These basic rights to access financial services can be a devastating weapon against a plural democracy where all views are allowed to exist. We can't really defend freedom of speech, if we aren't looking at the backdoor ways that we are having it attacked.}} ===Southern Poverty Law Center=== Graves noted, "You mentioned the [[w:Southern Poverty Law Center|Southern Poverty Law Center]] was attacked. My understanding of their purpose and their activities is to try to shut down the [[w:Ku Klux Klan|Ku Klux Klan]] and [[w:Neo-Nazism|neo-Nazi]] groups, and they've been moderately successful." Reitman agreed. {{quote| Yes. ... The basis of the indictment had to do with this idea that for a period of time, the Southern Poverty Law Center allegedly had a system where they were paying informants to find out more information about what was happening within the white supremacy movement. They might pay an informant to find out about upcoming events and then be able to prepare for them and do something about it. That program doesn't exist anymore, but it did for a while. The Trump administration is alleging that by having this paid informants program, they were somehow supporting white supremacy, which is a ridiculous idea. The indictment is ridiculous. Everything about that case is a little ridiculous, including because the Southern Poverty Law Center worked very closely with the police in developing that program, is my understanding. ... But the financial systems, these donor-advised funds that nonprofit organizations very much rely on to do their work, didn't wait for a judge to weigh in on it, and instead said "Just because there's an indictment, just because this is in the news, we are going to stop donations to the Southern Poverty Law Center." And I want to flag that they didn't have to do that. And in fact, there were other donor-advised funds, like the [[w:San Francisco Foundation|San Francisco Foundation]], that said, we absolutely are going to continue to allow donations to flow to the Southern Poverty Law Center ... .}} === Trend === Graves asked, "Has this become more common in recent years, since the year 2000, or since Trump's second inauguration?" Reitman replied, {{quote| I don't think we have enough data to answer that question. ... We would have to require companies to be more transparent about when they're shutting off accounts and why they're shutting off accounts. Right now, they don't have to put out any kind of reports about that, not to the public and not to Congress. I think we need more transparency.<ref>In some cases banks are prohibited by law from giving the real reason for freezing an account, as discussed in the section on "[[Rainey Reitman on how finance companies punish speech#Frozen accounts restrict withdrawals but allow deposits|Frozen accounts restrict withdrawals but allow deposits]]" under [[Rainey Reitman on how finance companies punish speech#Discussion|Discussion]] below.</ref>}} === How to respond? === Graves then noted, "I'm Secretary of PeaceWorks Kansas City,<ref><!--PeaceWorks Kansas City-->{{cite Q|Q64287449}}</ref> and I'm writing research reports to raise questions about the US military budget and priorities. I can imagine that I would be targeted and that Peaceworks would be targeted. What should we do if we are?" Reitman replied, {{quote| I don't have a lot of examples. I don't have a lot of specific advice for individual consumers. I did look at a couple of alternatives to using the financial system. I looked at cash. Some people are like, "Should I just be dealing with cash?" Cash is quite censorship-resistant, and it's much more difficult to surveil, so there's a lot of benefits there. But cash is very difficult if you're trying to process online transactions, which a lot of organizations need to do. And cash is quite bulky if you're trying to move it around in large quantities. If you're, say, a dental office, that's physically quite a lot of cash to manage. It's just not gonna be a great solution. I mention in my book, cash is also under attack. A lot of people are promoting the elimination or the reduction of cash in our society, that we would move to almost all digital transactions. That would have so much more surveillance, so much more control, and all these problems of censorship. So, rather than seeing cash as a savior, I really said cash is under attack right now. If we want to have it for future generations, we need to protect it. I also looked at cryptocurrency. Cryptocurrency is a very hot button issue. People have strong feelings about it. I didn't want to wade into all that. I just wanted this one question answered, which is, "Is it censorship resistant?" Could you use cryptocurrency if PayPal or Visa or whoever had shut you down? And what I said is, technically, yes, the protocol is pretty censorship resistant. But most people who are using cryptocurrency today are using it through a cryptocurrency company. They're using a [[w:Coinbase|Coinbase]] or [[w:Kraken (cryptocurrency exchange)|Kraken]] or one of these big cryptocurrency companies. And as soon as they do that, they've basically recreated the same systems of control that we have in the traditional financial world.}} ===Freedom of the Press Foundation=== Graves asked, "Talk about Freedom of the Press Foundation." Reitman replied, {{quote| Freedom of the Press Foundation is a non-profit that I co-founded with my friend Trevor Timm<ref><!--Trevor Timm-->{{cite Q|Q141606284}}</ref> back in 2012. Freedom of the Press Foundation works to promote freedom of speech and defend the rights of journalists in the United States and a little bit outside of the United States, increasingly. We do advocacy around the law, but we also do technical projects to provide tools for journalists. For example, we have a project called [[w:SecureDrop|SecureDrop]], which we have set up in all these newsrooms around the world that lets any news organization securely receive anonymous documents from whistleblowers. And then, let's say the government comes to those news organizations in the future and says, "You have to tell us where you got this." The news organization can say, "We don't know. We never got their information. It's a super secure system, so there's no way for us to find out." We maintain that system and a number of other projects that help support journalists. Maybe more broadly, I think we're a voice for unyielding support for the First Amendment here in the United States, which I think is incredibly important at this moment.}} === Penlight prize === Reitman continued, {{quote| We have a new prize: The Pennlight Prize is a project that we launched with the support of a donor this year. The idea is to provide a $25,000 prize to a journalist who publishes an article based on [[w:Freedom of Information Act|Freedom of Information Act]] documents that they published outside of a paywall. Here's why we're doing that. One of the things that has happened, and I completely understand why it's happening, is that so much important and necessary information that the public needs to know is increasingly behind paywalls. There are so many different news organizations that people don't end up getting a subscription to 55 different news organizations. So they hit a paywall and they leave. And where do they go? They go to less reliable news organizations. They go to misinformation websites. They get bogus data because it's not behind paywalls. And they're filled up with a whole bunch of wrong data. So we wanted to find a way to encourage at least a tiny amount of information, super high-quality journalism, to be outside of paywalls and to help inform the public. And Freedom of Information Act requests are documents based on information that we have a right to know. It was made publicly available, so we want the underlying documents to be made public, and we want the articles about those documents to also be outside of a paywall. So we made this a very big journalism prize. Journalists traditionally don't make a lot of money. Our hope was that by making it a significant prize, $25,000, that it would incentivize journalists to really advocate for these policies internally within their news organizations. This is the first year we're doing it. We had over 100 submissions. We're working through them right now. My hope is that next year, maybe we get 300 submissions, and we get more news organizations saying, "We're taking these important articles. We're putting them outside of the paywall, so that more people can get access to them."}} === Changes since publication? === Graves asked, "What has changed in this regard since your book went to press?" Reitman replied, "I submitted this book before I knew if Kamala or Trump was going to win, and I really had no idea what kind of a political reality this book was going to be born into. ..." Graves asked, "What reactions have you gotten from the book?" Reitman replied, {{quote| Very positive for the most part from the civil liberties community and from lots of different folks in the tech sphere. The financial companies have been less positive about it. Some of the feedback I got from people in the financial space was this idea that they're the victims, right? "We are just doing what we have to, and you're painting us as the villains", which I feel is overly simplistic and frankly not accurate. I think there's a lot of power within these financial companies to develop positions that could harm or hurt consumers, and they're not doing their due diligence to stand up for free speech ... .}} Graves asked, "What about elected officials like Senator Elizabeth Warren?" Reitman replied, "I haven't talked to Senator Elizabeth Warren about it." === Final words === When asked for final words for the audience, Reitman replied, "If folks are interested, one of the best ways to support a new author is to buy a book from a local bookseller or whoever. It's called ''Transaction Denied.'' And half of the author proceeds go to support the Freedom of the Press Foundation ... ." Graves added that he had previously interviewed [[Freedom of the Press Foundation says...|a couple of the leaders from Freedom of Press Foundation]] earlier in this Media Democracy series, "and I've been a long-time supporter." ==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/>]'' ===Frozen accounts restrict withdrawals but allow deposits=== In this interview Graves asked, "Can they confiscate my Social Security or retirement annuities?" Reitman replied, "As far as I know, they cannot." Further research suggests the reality is complicated in part, because accounts can be frozen in some cases by creditors as well as under anti–money laundering laws. Eneriz (2026) says, "A frozen bank account restricts you from withdrawing or transferring funds but allows deposits." fedlaws.org (2026) provides more detail with legal citations. They say, {{quote| A freeze blocks withdrawals, transfers, and scheduled payments while keeping the account technically open. ... When a bank’s monitoring flags a suspicious pattern, it files a Suspicious Activity Report with the Financial Crimes Enforcement Network. ... :''These freezes are the most disorienting because the bank is legally prohibited from telling you the real reason.'' (emphasis added) ... Even when a garnishment order is perfectly valid, certain money in your account is off-limits. Federal law protects direct-deposited benefits from Social Security [and some other funds]. ... When a bank receives a garnishment order, it must look back over the prior two months of deposits and calculate a “protected amount” equal to the total federal benefit payments deposited during that window. The bank must give you full access to that protected amount without requiring paperwork or an exemption claim. ... If your account contains additional exempt funds above the protected amount, you can assert a further exemption by contacting the creditor, completing an exemption form with the court, or working with a legal aid attorney. One detail most people miss: direct deposit matters. If you receive benefits by paper check and deposit them yourself, the automatic two-month lookback may not identify the funds as protected. Direct deposit creates the electronic trail the bank’s system uses. Joint Accounts Put Your Money at Risk ... The law generally presumes each co-owner has equal rights to every dollar in the account, so a creditor pursuing one account holder can freeze the entire balance. .. A non-debtor co-owner can fight this, but must act fast. The garnishment notice includes a hearing date, and requesting that hearing in writing before the deadline is critical. ... Miss the hearing and the court can release the entire balance to the creditor.}} === ''The Chickenshit Club'' === The interview mentioned potential fines of over $360,000 per violation, mentioned by Reitman (2026, p. 60). A web search for documentation of such fines identified Moss (2026), "TD Bank to Pay Record $3 Billion for [[w:Bank Secrecy Act|BSA]]/AML Violations". However, no mention was found of fines "over $360,000 per violation." $3 billion may sound like a lot of money but may be a minor cost of doing business for a large organization like [[w:TD Bank (United States)|TD Bank]]. The article by Moss (2026) reported that, {{quote| senior management at TD Bank knew that its failure to adequately support [[w:Anti–money laundering|AML]] compliance created backlogs that allowed illicit transactions to continue to flow through the bank. ... Notably, the bank: ... Allowed trillions of dollars of domestic [[w:ACH Network|ACH]], remote deposit capture, and peer-to-peer (P2P) transactions to go unmonitored for over a decade." It's not immediately obvious how much money TD Bank made from having "allowed trillions of dollars of ... transactions to go unmonitored". Might $3 billion have been a small percentage of what they made violating the law? The fact that the settlement did not include prison time for any "senior management" is consistent with the discussion in "[[Dean Starkman and the watchdog that didn't bark]]" of "Whitewashing corporate crime ... described by Eisinger (2017) :[''The chickenshit club : why the Justice Department fails to prosecute executives'']. :''When these deals are announced with the Justice Department, invariably the shares of the company go up, not down.'' (emphasis added)}} == Notes == {{reflist}} == Bibliography == * <!--Jesse Eisinger (2017) The chickenshit club : why the Justice Department fails to prosecute executives-->{{cite Q|Q134599351}} * <!--Ashley Eneriz (2026-09-25) "Frozen Bank Account: Common Causes and Steps to Unfreeze"-->{{cite Q|Q141599984}} * <!--fedlaws.org (2026-09-23) "Why Do Banks Freeze Accounts? Triggers, Exemptions, and Fixes-->{{cite Q|Q141599719}} * <!--Valerie Moss (2024-10-17) "TD Bank to Pay Record $3 Billion for BSA/AML Violations"-->{{cite Q|Q141600174}} * <!--Rainey Reitman (2026-04-07) Transaction Denied: Big Finance's Power to Punish Speech-->{{cite Q|Q141526890}} [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Political economy]] [[Category:Freedom of speech]] [[Category:Finance]] [[Category:Economics and Personal Finance]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> 7zxm1w8j0mt112dyy0o1ejaqc1zgx2f 0 332288 2835557 2835397 2026-10-02T19:26:06Z Scogdill 1331941 2835557 wikitext text/x-wiki = Wedding of Miss Clarita de Murietta and the Duc de Santona = ==Overview== This wedding was between Spanish Catholics, but it took place in England. The de Murriettas were friends of the Prince of Wales, and many people from his social network in 1890 were present at this wedding. The Murrietas lived in the UK (in London, with a country estate in Surrey), but the Duc de Santona lived in Spain. The ''Lady's Pictorial'' covered this wedding thoroughly, including drawings of some of the bride's dresses.<ref>"Santona—De Murrieta." ''Lady's Pictorial'' 1 February 1890, 30 [of 60], Col. 1a [of 3] – 31, Col. 1a–2a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0005980%2F18900201&page=30 – . Same print title, pp. 158–159.</ref> ==Logistics== * The wedding took place at 11:30 on Wednesday, 29 January 1890 at Brompton Oratory. People invited to the church ceremony had tickets. The reception seems to have been a breakfast, but after the ceremony. * "The weather was all that could desired of a winter's day, presenting a bright and cheerful contrast to the storms which have lately proved so destructive and depressing. That is to say, the wind had moderated, the sun shone brightly, and the air was tinged with exhiliarating [sic] crispness. As early ten o'clock — an hour and half before the ceremony — the holders of tickets for admission to the sacred building began to arrive, and crowd to assemble round the gateways. At eleven every seat, except those reserved for the wedding party, was occupied, and from that time till nearly one o’clock a long line of carriages occupied both sides of the broad roadway from the Oratory to beyond South Kensington Museum."<ref name=":2">"Marriage of Miss de Murrieta to the Duke of Santona." ''Sussex Express'' 1 February 1890, Saturday: 7 [of 12], Cols. 1a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0000654%2F18900201&page=7&article=136&stringtohighlight=mandeville+viscount. Print title: ''The Sussex Express, Surrey Standard, Weald of Kent Mail, Hants and County Advertiser'', p. 7.</ref> ===Staff and Vendors=== *"The Bishop of Southwark, the Rev. Canon Searle (of Tunbridge Wells), the Rev. Father Gordon, of the Oratory, who performed the ceremony, entered the church in procession, with their attendants."<ref name=":2" /> *"An orchestra of nearly 40 string instruments, with the organ, performed for three-quarters of hour before the time fixed, 11.30."<ref name=":3" /> *"The music was as follows:— Before service, slow movement of minuet and trio for Italian Symphony (Mendelssohn) and Allegretto from 8th Symphony (Beethoven). As the bride entered the church, 'Bridal march' from Lohengrin. While the register was signed, offertoire from 'Messe Solennelle.' During Mass, Gounod's 'Mess Solennelle,' offertory, Gounod's 'Ave Maria,' solo by Master Hawkins. Finale, Mendelssohn’s 'Wedding March.'"<ref name=":2" /> ==Related Events== * The ''Bedfordshire Mercury'' reported that the Marques and Marquesa de Santurce hosted some notables at their London residence, 18, Carlton House-terrace.<ref name=":3" /> * Reception * Honeymoon ==Who Was Present== ===Bride and Bridesmaids=== ====Bride==== "Clarita de Murrieta, younger daughter of the Marques and Marquesa de Santurce, of Wadhurst Park, Sussex, and 18, Carlton House-terrace"<ref name=":2" /> ==== Bridesmaids ==== * Mdlle. Marie de Mitjans (sister of the bridegroom) * The Hon. Muriel Agnes Brassey (daughters of Lord Brassey, of Normanhurst) * The Hon. Marie Adelaide Brassey * Miss Joan Marion Nevill (daughter of Lord Henry Nevill) * Miss Violet Gathorne-Hardy (daughter of the Hon. C. C. and Lady Cicely Louisa Gathorne-Hardy) * Miss Constance Barron (daughter of Capt. and Mrs. Barron)<ref name=":3" /> ====Pages==== ===Groom and Best Man=== * "Juan Manuel Mitjans, Duque de Santona, a grandee of Spain"<ref name=":2" /> * "Mr Francis de Murrieta, cousin of the bride, was best man."<ref name=":3" /> ===People Who Attended=== "The company present at the wedding ceremony, and afterwards at the breakfast at Carlton House-terrace, included,"<ref name=":2" /> # The Prince of Wales # the Marquesa de Manzanedo and Mdlle. Maria Mitjans # the Duque and Duquesa de Lecera # the Duchess of Leinster and Lady Helen Duncombe # the Duchess of Manchester # the Dowager Countess of Airlie and Lady Griselda Ogilvy # [[Social Victorians/People/Mandeville|Viscountess Mandeville and Lord Kimbolton]] # Viscountess Hood # Lord and Lady Brooke and Lady Sybil St. Clair-Erskine # Lord and Lady William Nevill # Lord Brassey # Lord and Lady George Nevill # Hon. Thomas and Lady Idina Brassey # Lord and Lady Henry Nevill # the Bishop of Southwark # Lady Frances Pratt and Lady Clementine Pratt # Lady Lister-Kaye # Lady Dorothy Nevill # Lady Cicely Gathorne-Hardy # Lady Sandhurst # Lady Alice Morland # Lord Richard Nevill # Sir Frederick and Lady Milner # Hon. C. N. Spencer, M.P., and Mrs. Spencer # Mr. and Lady Margaret Douglas # Hon. Humphrey Sturt # Sir Oscar Clayton # Lady Constance Leslie and Miss Leslie # Sir William Gordon-Cumming # Hon. Mrs. Agar-Ellis and Miss Ellis # Mons. and Madame Escandon and Mdlle. Escandon # M. Calderon # M. Errazu # M. Galintz # Mr. Frank Lee # Major Swaine # Mr. and Mrs. Granville Ward # M. de Soveral # Mr. W. H. Mitchell # the Rev. Mr. Palmer # Mr. C. de Murrieta # Mr. A. de Murrieta # Mr. Mariano de Murrieta # Mr. C. J. de Murrieta # Mr. and Mrs. Barron # Mr. Hatch # Mr. Reginald Ronalds # Mrs. Temple Soanes # Dr. and Mrs. Robson Roose # Mr. and Mrs M. Van Raalte # Mr. Grenfell # Mrs. Bischoffsheim # Mr. Hulse, M.P., and Mrs. Hulse # Mr. Wood # Dr. White # Mrs. Ronalds and Mrs. Ritchie # Mr. Leo Schuster # Mrs. Ince Anderton # Mr. Eldrid # Mr. and Mrs. Hammond # Mr. and Mrs. E. H. Eldrid # the Rev. J. S. Welby # Mr. F. O. Smithers # Mr. and Mrs. Moreton Frewen # Mr. and Mrs. Granville Ward # Mrs. and Miss Silvertop # Mr. Gerald Paget # Mr. and Mrs. Hwfa Williams # Mr. Charles Van Raalte # Mr. Albert Stopford # Hon. Mrs. Vaughan # Miss E. Vaughan # Miss Townshend # Mrs. Coupland # Mrs. and Miss Leigh # Mr. and Mrs. F. V. Williams # the Spanish Consul-General ==What People Wore== <blockquote>The Bride. — The bride’s dress was of rich white Venetian brocade, with a very long court train; the front was bordered round the hem with fringes and bouquets of orange blossom, a garland of the same being arranged on the left side. The bodice had a high collar of delicate lace and a softly-draped fichu of lace and fine muslin. She wore small crown of orange-flowers and a tulle veil fastened to the hair with diamond ornaments and carried a lovely bouquet of exotics. The Bridesmaids. — The bridesmaids’ pretty costumes were of soft cream crepon, with sleeves and vests of apple green velvet; the straight skirts were edged with a single row of gold braid; and they wore sashes of pale mauve moiré, the ends of which fell from the waist in front to the feet. The two elder bridesmaids wore small bonnets of green velvet and gold braid, and the other four picturesque hats match, trimmed with the braid and several green ostrich feathers. The bridegroom's gift to each bridesmaid was also worn. It was a very handsome brooch formed of a horseshoe in diamonds and sapphires pierced by a diamond arrow; in the centre of the horseshoe were the initials "J. C." in diamonds, with a ducal coronet above in rubies and diamonds. The bridesmaids carried bouquets of lilies of the valley and green orchids, tied with mauve and green ribbons. The Bride’s Mother. — The Marquesa de Santurce's costume was entirely of dark reseda velvet, with the exception of the sleeves, which were of embossed silk with a design of foliage delicately outlined in gold. The straight collar was also of embossed brocade. The Marquesa’s bonnet was composed of two wing-like arrangements of reseda-green lisse, wrought with lines of gold, placed eave-like over the wearer's dark hair. The crown, a very tiny one, almost hidden by the wings, was of velvet embroidered in gold. The bouquet carried was an exquisite combination of white orchids, Maréchal Niel roses, and mimosa. It was tied with ribbon of the same colour as the costume. The Bridegroom's Mother. — The Marquesa de Manzanedo, the bridegroom’s mother, wore train of almost courtly proportions. It, as well as the bodice, was of dark heliotrope velvet. The bodice was made with puffed sleeves, very high on the shoulders, and a Medici collar This was faced, and the bodice bordered, with Russian sable. The skirt was of pearl-coloured brocade, backed with plaler heliotrope. It was deeply flounced with dark rich sable tail, and over it at one side a scarf of priceless Spanish lace was draped. A bonnet of lisse and velvet in corresponding tones of heliotrope colour was worn. The Marquesa carried a magnificent bouquet of orchids of the same shade her costume, arranged with varied foliage and tied with dark heliotrope ribbon. The Bridegroom's Sister.—The Duquesa de Lecera, the bridegroom's sister, was attired in a striking costume of pale pink satin striped with black. It was princess in form the stripes symetrically converging towards the waist. The skirt was bordered with soft ruche of pale pink chiffon. A pretty little bonnet composed of lisse and velvet matched the dress, and the Duquesa carried a bouquet of pale pink carnations and orchids. The Bride’s Sister.—Lady William Nevill’s dress was of Sicilienne and velvet. The form was of the polonaise order and the colour serpent-green. The skirt was quite plainly arranged, and entirely of Sicilienne. Lady William’s bonnet was of velvet, a ruche of pale pink China crêpe showing above the hair. Pink and green ospreys were placed at one side. Her bouquet was mass of pink blossoms arranged with fern fronds, and tied with serpent-green ribbon. Lady Cicely Gathorne-Hardy's dress was a pretty combination of electric-blue velvet and cashmere. There was soft vest of white China crêpe. The bonnet was of velvet, similar that of the dress; a grey and white bird was coquettishly placed at one side. Lady Cicely’s bouquet was of white roses and pink carnations. Lady Henry Nevill’s dress was made of light cloth of a pale tone of fawn-colour striped with narrow lines of brown. The sleeves and top portion of the bodice were of violet velvet. pointed panel of similar velvet showed at either side of the skirt. The cloth was drawn in folds  over the shoulders, and was caught under folds of velvet at the waist, showing deep V-shaped points of the bright-hued velvet in front and at the back. A bonnet of fawn-coloured silken gauze was covered with violet velvet pansies laid flatly on the crown. Lady Henry's bouquet was of shaded violets. Lady George Nevill wore a long mantle of dark-brown cloth broché with red. The sleeves, made high at shoulders were of dark autumnal-brown velvet, and the bonnet was of velvet, of the same colour, amid loops of which dark-red plumaged birds were nestled. Lady Idina Brassey's costume of ruby velvet was very effective, being flounced and bordered with dark fur. The hat was of felt, matching the dress in colour, and its flat crown was surrounded with dark red ostrich feathers. Lady Rose Leigh was quietly dressed in a neat tailor-made tweed gown, dark blue in colour, intercepted by narrow lines of white. A hat of dark blue felt, bordered with cock’s feathers twisted into a curl over the flat crown, looked appropriate to the studied simplicity of the dress. Lady Dorothy Nevill was attired in a long mantle of wall-flower red plush trimmed with bronze passementerie. Her headgear was a neat, closely-fitting bonnet, matching her mantle in color. The Duchess of Manchester wore a dress of silver-grey silk brocaded with satin; a section of plain satin, arranged in straight pleats, formed the back. Her Grace's mantle was of velvet — of so dark a shade of [Col. 1c–2a] myrtle green as to appear black in the shadow — and it was trimmed with lines of jet embroidery. The bonnet worn by the Duchess was of sinrlar velvet, with a spray of jet across the front. The Duchess of Leinster, who was accompanied by her sister, Lady Helen Duncombe, was attired in golden brown cloth trimmed with a design of foliage in velvet, applied to the surface of the cloth, with silk embroidery. Her Grace’s coat was of embossed silk of a delicate shade of stone colour slightly wrought with gold. The pretty but very small hat worn by the duchess was of brown felt, and her Grace's boa of ostrich feathers shaded from brown to pale stone colour. Lady Helen’s dress was of dark reddish brown woollen material, simple in form but of perfect style. Her hat was also of felt, matching her costume in colour. Lady Algernon Gordon Lennox. — A strikingly-handsome costume was that worn by Lady Algernon Gordon-Lennox. It was of bright soft-red velvet trimmed with bands of red-gold embroidery. Her bonnet was a very small one of red velvet, with a spray of ferns  simulated in jet across the front. Lady Algernon wore a long boa of sable-tail. The Dowager Countess Airlie. — Was costumed in her invariable sable robes of almost cloister-like severity, which so admirably suit her distinguished figure and beautiful face. Lady Grisel [Griselda?] Ogilvy. — Wore a black skirt, with a broad sash of pale blue moire. Her cape was of fine steel beadwork, and she wore a Henry VIII. hat of black velvet, with a group of pale blue feathers at one side. Mrs. Ronalds. — Wore a dress and mantle of black velvet bordered with sable tail, and a bonnet ''en suite''. Her daughter, Mrs. Ritchie. — Was gowned in willow-green Royal silk, and wore a bonnet of green lisse and mantle made entirely of jet beadwork.<ref name=":2" /></blockquote>The bride's going-away dress: "The bride went away in a costume of grey velvet, of a delicate shade, draped with ''Sicilienne'', and trimmed with rich steel passementerie; a broad band  of grey velvet edged the skirt, and she wore a jacket and hat to match, and sable boa and muff."<ref name=":2" /> ==Gifts== <blockquote>THE WEDDING PRESENTS Were displayed in the ball-room, at the house of the Marquess De Santurce in Carlton House Terrace. The jewellery was simply magnificent, and the others were also costly and elegant. The presents to bride and bridegroom numbered about 500, including the following:—<ref name=":2" /> [set as unordered list, originally set as separate paragraphs] TO THE BRIDE * The P<small>RINCE</small> and P<small>RINCESS</small> of W<small>ALES</small>, moonstone heart brooch, set with rubies and diamonds. * Princess L<small>OUISE</small>, Duchess of Fife, and the Duke of F<small>IFE</small>, a set of silver salt-cellars. * The Empress E<small>UGENIE</small>, a large silver mirror. * The B<small>RIDEGROOM</small>, a magnificent sapphire and diamond necklace, a sapphire and diamond bracelet, ruby and diamond ring, pearl double-heart brooch, diamond pin, diamond and pearl brooch, gold and pearl studs, and gold-handled hunting crop. * The Marquess and Marquesa <small>DE</small> S<small>ANTRUCE</small> [sic], a magnificent diamond tiara. * Marquesa <small>DE</small> S<small>ANTURCE</small>, diamond half-hoop ring. * M. M<small>ITJANS</small>, father of the bridgegroom, three rows of large pearls, with diamond and emerald clasp. * Marquesa De M<small>ANZANEDO</small>, mother of bridgeroom [sic], ruby and diamond cluster necklace. * Lord and Lady W<small>ILLIAM</small> N<small>EVILL</small>, handsome dressing bag with tortoiseshell and gold fittings, and enamel and diamond watch bracelet. * Mr. C<small>HRISTOBAL</small> and Mr. A<small>DRIANO DE</small> M<small>URRIETA</small>, diamond aigrette, and two pairs of carriage horses. * Mr. M<small>URIANO DE</small> M<small>URRIETA</small>, silver-gilt tea service and tray, three silver-gilt candelabras, and set of silver-gilt dessert plates. * The Duc and Duchesse <small>DE</small> V<small>ISTAHAMOSA</small>, complete set of silver-gilt dessert forks, knives, and spoons, and four pairs of old English silver candlesticks. * Mr. F<small>RANCIS</small> and Mr. C. J. <small>DE</small> M<small>URRIETA</small>, pair of large silver-gilt repoussé dishes. * Marques <small>DE LA</small> T<small>ORREVILLE</small>, diamond double-heart  brooch. * M. C<small>ALDERON</small>, set of diamond hair pins. * Marchioness of A<small>BERGAVENNY</small>, black point lace and tortoiseshell fan, with monogram in rubies and diamonds. * Duke and Duchess of N<small>EWCASTLE</small>, diamond and emerald watch-bracelet. * Mr. A<small>LFRED DE</small> R<small>OTHSCHILD</small>, diamond and pearl bracelet. * Lord and Lady S<small>ANDHURST</small>, gold "1890" sleevelinks. * Mdlle. M<small>ARIA</small> M<small>ITJANS</small>, large carbuncle and diamond pendant. * The S<small>PANISH</small> A<small>MBASSADOR</small>, six silver coffee cups. * Duchess of M<small>ANCHESTER</small>, pair of silver-gilt-topped scent bottles. * Duchess of L<small>EINSTER</small>, old china box. * Hon. R<small>OBERT</small> and Mrs. S<small>PENCER</small>, diamond pin. * Marquis of A<small>BERGAVENNY</small>, writing set and candlestick. * Countess of D<small>ENBIGH</small>, embroidered silk blotting book. * Lady A<small>LICE</small> M<small>ORLAND</small>, silver vase. * Lord R<small>ICHARD</small> N<small>EVILL</small>, pigskin luncheon bag with complete silver fittings. * Hon. T<small>HOMAS</small> and Lady I<small>DINA</small> B<small>RASSEY</small>, leather carriage card case, with gold mountings and monogram. * Lord and Lady B<small>ROOKE</small>, Dutch silver ornament. * Viscount and Lady V<small>IOLET</small> D<small>ANGAN</small>, old painted snuff-box. * Sir O<small>SCAR</small> C<small>LAYTON</small>, pair of silver ash trays. * Sir W<small>ILLIAM</small> G<small>ORDON</small> C<small>UMMING</small>, silver-headed scent bottle. * Viscountess M<small>ANDEVILLE</small>, small green leather travelling bag. * Mrs. C<small>OUPLAND</small>, silver crucifix. * Canon S<small>EARLE</small>, benetier. * Lady B<small>ROUGHAM</small> and V<small>AUX</small>, old gold-mounted scent  bottle. * Lady C<small>LIFFORD</small> and Miss T<small>OWNELEY</small>, silver looking-glass. * Lord A<small>SHBURTON</small>, silver box. * Lady A<small>LGERNON</small> G<small>ORDON</small>-L<small>ENNOX</small>, inlaid tortoiseshell box. * Dowager Duchess of N<small>EWCASTLE</small>, silver tray. * Lord and Lady C<small>HARLES</small> B<small>ERESFORD</small>, small Louis XVI table. * Sir F<small>REDERICK</small> and Lady M<small>ILNER</small>, silver tray. * Sir Bach Cunard, scent bottle with gold top set with diamonds. * Mrs. G. C<small>AVENDISH</small>-B<small>ENTINCK</small>, pair of small old silver vases. * Mr. and Mrs. E<small>DWARD</small> S<small>ASSOON</small>, Sevres china ink-stand. * Mr. J. B. and Lady R<small>OSE</small> L<small>EIGH</small>, Chippendale table. * Lord and Lady H<small>ENRY</small> N<small>EVILL</small>, complete silver tea set. * Mr. H<small>ORACE</small> F<small>ARQUHAR</small>, two girandoles. * Colonel R<small>ALPH</small> V<small>IVIAN</small>, umbrella with gold handle. * T<small>HE</small> S<small>ERVANTS AT</small> W<small>ADHURST</small> PA<small>RK AND</small> 18, C<small>ARLTON</small> H<small>OUSE-TERRACE</small>, large silver salver. * T<small>HE</small> W<small>ADHURST</small> T<small>ENANTS AND</small> T<small>RADESMEN</small>, large silver cup. * W<small>ADHURST</small> P<small>ARK</small> S<small>TABLEMEN</small>, driving whip and hunting crop. * Marquis C<small>AMDEN</small> and Lady C<small>LEMENTINE</small> P<small>RATT</small>, silver box. * Mr. R<small>ICARDO DE LA</small> Q<small>UININA</small>, Minton china tea and coffee set and dinner service. * Mr. H<small>EREDIM</small>, diamond and enamel watch bracelet. * Duque and Duquesa <small>DE</small> L<small>ACERA</small>, diamond and sapphire brooch. * Mr. and Mrs. D. Y<small>BARROMDO</small>, large silver bowl. [Col. 2c–3a] * Mr. F. V. and Miss W<small>ILLIAMS</small>, four silver-gilt salt cellars. * Miss L<small>EIGH</small>, leather letter holders. * Duqe and Duquesa <small>DE</small> F<small>ERNAN</small>-N<small>UNEZ</small>, old Spanish fan. * Duque and Duquesa <small>DE</small> A<small>LBA</small>, old Spanish fan. * Mrs. and the Misses C<small>ARROLL</small>, silver hand-glass. * Lord George N<small>EVILL</small>, hunting crop. * Lady George N<small>EVILL</small>, silver photo frame. TO THE BRIDEGROOM. * The Bride, turquoise and diamond pin, and ruby and diamond ring, and Cabochen sapphire and diamond sleeve links. * The Marquesa <small>DE</small> M<small>ANZANEDO</small>, emerald and diamond  pin. * The Marquesa <small>DE</small> S<small>ANTURCE</small>, a brougham, sapphire and diamond pin, and ruby and diamond horseshoe pin. * Mr. C. and Mr. A. <small>DE</small> M<small>URRIETA</small>, pair of guns. * Lord and Lady W<small>ILLIAM</small> N<small>EVILL</small>, set of crystal and silver liqueur cups and bottles on crystal and silver tray. * Mr. F<small>RANCIS DE</small> M<small>URRIETA</small>, silver looking-glass. * Mr. C. J. <small>DE</small> M<small>URRIETA</small>, pearl pin. * Count T<small>ORRE</small>-A<small>RIAS</small>, silver-mounted paper knife. * Comte <small>DE LA</small> P<small>ATELLA</small>, pearl pin.   * Count <small>DE</small> B<small>ENATUA</small>, Rahi [?] cart. * The Duque and Duquesa <small>DE</small> L<small>ECERA</small>, gold links and studs. * Duc <small>DE</small> A<small>LBA</small>, phaeton. * Mr. J. G<small>RENFELL</small>, silver cigarette case. * Mr. T<small>HOMAS</small> W<small>OOD</small>, umbrella and cigar case. * Lord R<small>ICHARD</small> N<small>EVILL</small>, leather box, containing cards, counters, &c. * Mr. P. <small>DE</small> G<small>ALINDEZ</small>, silver cigar case. * Madame <small>DE LOS</small> H<small>EROS</small>, tea basket. * Duchesse <small>DE</small> A<small>LIAGA</small>, set of books. * M. and Madame E<small>SCAUDON</small>, sapphire and diamond pin. * Count E<small>STRADOS</small>, gold match-box. * M. R<small>AMON DE</small> E<small>RRAZU</small>, silver-fitted dressing bag. * M. L<small>UTS DE</small> E<small>RRAZU</small>, diamond pin. * Sir W<small>ILLIAM</small> G<small>ORDON</small>-C<small>UMMING</small>, old silver box. * The S<small>ERVANTS</small> at Wadhurst and 18, Carlton House-terrace, pair of silver-mounted riding whips.<ref name=":2" /> </blockquote> ===Unusual or Interesting Gifts=== ==Anthology== The Marques and Marquesa of Santurce hosted the wedding and reception of their daughter, Miss Clarita de Murietta, with the Duc de Santona. ==== The House Party ==== This ''Bedfordshire Mercury'' reports on the wedding and what seem to be local celebrities. Given the focus on [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess of Manchester]] and her daughter-in-law [[Social Victorians/People/Mandeville|Consuelo, Lady Mandeville]], the ''Bedfordshire Mercury'' looks like a paper local to them.<blockquote>The Duchess of Manchester, Vicountess Mandeville and Lord Kimbolton were among the guests entertained by the Marques and Marquesa de Santurce at their residence, Carlton House-terrace, London, on the occasion of the marriage of their younger daughter, Mdlle. Clarita de Murrieta, with the Due de Santona, last Wednesday. The ceremony took place at Brompton Oratory, where the floral decorations were most elaborate, whole trees of white lilac and camelias, besides a profusion of arums and lilies of the valley, being artistically arranged about the sanctuary and in adjacent chapels. An orchestra of nearly 40 string instruments, with the organ, performed for three-quarters of hour before the time fixed, 11.30. The bridesmaids were Mdlle. de Mitjans (sister of the bridegroom), the Hon. Muriel and Hon. Marie Brassey, Miss Joan Nevill, Miss Violet Gathorne-Hardy, and Miss Constance Barron. Mr Francis de Murrieta, cousin of the bride, was best man. The Bishop of Southwark performed the nuptial rite, and afterwards said Mass, assisted by the Rev. W. F. Gordon, and other Fathers; Gounod’s "Messe Solennelle" was sung by nearly 40 voices, and his "Ave Maria," adapted from Bach, by a Master Hawkins. The Prince of Wales, the Duchess of Leinster and Lady Helen Duncombe, Viscountess Hood, Lady Frances and Lady Clementine Pratt, the Hon. C. R. and Mrs Spencer, Mr Grenfell, Mr and Mrs Hulse, &c., were also of the bridal company. The Duchess of Manchester wore at the church a long velvet mantle trimmed with sable, which almost concealed her grey dress. The donors of wedding presents included the Empress Eugénie, the Prince and Princess of Wales, the Duke and (Princess Louise) Duchess of Fife, Mr Alfred Rothschild, &c. The Duchess of Manchester gave the bride a pair of silver-gilt-topped scent bottles, and Lady Mandeville a small green leather travelling bag. The Duc de Santona and his bride left town for Wadhurst Park.<ref name=":3">"Kimbolton." ''Bedfordshire Mercury'' 1 February 1890, Saturday: 8 [of 8], Col. 3b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0001290%2F18900201&page=8&article=101&stringtohighlight=mandeville+viscount. Same print title and p.</ref></blockquote> ==== The Wedding ==== This report from the ''The Sussex Express, Surrey Standard, Weald of Kent Mail, Hants and County Advertiser'' takes up about 2 1/3 full columns:<blockquote>The marriage was solemnised, on Wednesday, at Brompton Oratory, with imposing ceremonial, and under most auspicious circumstances, of Juan Manuel Mitjans, Duque de Santona, a grandee of Spain, to Clarita de Murrieta, younger daughter of the Marques and Marquesa de Santurce, of Wadhurst Park, Sussex, and 18, Carlton House-terrace. The [happy event?] we now record not only attracted a numerous and brilliant congregation of the fashionable world to the service at the Oratory and the subsequent festivities, but was [invested?] with considerable interest to the wide borders of Kent and Sussex, from the sea to beyond the Weald, the bride being connected by the marriage of her sister to Lord William Nevill, last year, with the noble local families of Nevill, Gathorne-Hardy, Brassey, and Egerton. Her father, too, the Marques de Santurce, is now quite a Sussex man, having for years past been seated at Wadhurst Park. The weather was all that could desired of a winter's day, presenting a bright and cheerful contrast to the storms which have lately proved so destructive and depressing. That is to say, the wind had moderated, the sun shone brightly, and the air was tinged with exhiliarating [sic] crispness. As early ten o'clock — an hour and half before the ceremony — the holders of tickets for admission to the sacred building began to arrive, and crowd to assemble round the gateways. At eleven every seat, except those reserved for the wedding party, was occupied, and from that time till nearly one o’clock a long line of carriages occupied both sides of the broad roadway from the Oratory to beyond South Kensington Museum. The interior of the church, long before the arrival of the bride, presented a brilliant appearance. Rays of sunshine from the clerestory windows brought into relief the beautiful floral decorations within the sanctuary. Groups of tall palms and camellia trees in blooms, interspersed with white lilac and arum lilies, were placed in front of the chancel, while on the fine marble balustrade were large brass bowls filled with white azaleas, lilies of the valley, and roses. Between the massive columns separating the transepts from the aisles, were also palms and ferns interspersed with rose-tinted camellias and other flowers. The altars were also richly decorated. On the high altar were eight large bouquets of choice white exotics. The altar of Our Lady in the East transept was decked in bouquets of white, orange, and a pale shade of yellow, while that of St. Philip the west transept, was arranged with palms and scarlet and white blooms. THE CEREMONY. The choral part of the service commenced at eleven o’clock, when the choir, numbering about one hundred performers, including trumpeters and stringed instrumentalists, commenced a programme of grand sacred music. The Duke de Santona, who was attended by the bride’s cousin, Mr. Francis Murrieta, as best man, arrived several minutes before the half-hour, and awaited, at a ''prie-Dieu'' placed in front of the chancel balustrades, the coming of his ''fiance'' [sic]. The bridesmaids reached the Oratory just afterwards in readiness to take their part in the bridal procession. They were Miss Marie Mitjans (sister of the bridegroom), Miss Joan Marion Nevill (daughter of Lord Henry Nevill), Miss Violet Gathorne-Hardy (daughter of the Hon. C. C. and Lady Cicely Louisa Gathorne-Hardy), Miss Barron (daughter of Capt. and Mrs. Barron), the Hon. Muriel Agnes Brassey,  and the Hon. Marie Adelaide Brassey (daughters of Lord Brassey, of Normanhurst). Then came the Prince of Wales, who had journeyed from Sandringham specially for the occasion. His Royal Highness, attended by  Colonel Teesdale, was conducted to a seat in front of the sanctuary. The Bishop of Southwark, the Rev. Canon Searle (of Tunbridge Wells), the Rev. Father Gordon, of the Oratory, who performed the ceremony, entered the church in procession, with their attendants, from the north door of the west transept, and took their position in front of the high altar. The bride was accompanied to church by her mother, and the Marques de Santurce received his daughter at the main entrance to the nave. The bridal procession was at once formed. The fair and stately bride — fairer than most of the daughters of the sunny south — was escorted by her father through the nave and transepts to the ''prie-Dieu'' in front of the entrance to the sanctuary, the bridesmaids following, and the instrumental part of the choir performing the spirited bridal match [sic] from Lohengrin. During the first few minutes of the holy ceremony, the bride and bridegroom remained in prayer in front of the chancel. They were then conducted to the altar, before which, the marriage service was celebrated. It occupied but a few minutes[.] The clergy, the bride and bridegroom, the Prince of Wales, the parents of the happy pair, and several of their nearest relatives repaired to the vestry to sign the register. The Prince gave his arm to the Marquesa de Santurce, and signed the register as one of the witnesses. On returning to the church Nuptial Mass was celebrated, and the whole service which occupied less than an hour, was brought to a conclusion by the Bishop bestowing the blessing of the Church upon the young couple. The music was as follows:— Before service, slow movement of minuet and trio for Italian Symphony (Mendelssohn) and Allegretto from 8th Symphony (Beethoven). As the bride entered the church, "Bridal march" from Lohengrin. While the register was signed, offertoire from "Messe Solennelle." During Mass, Gounod's "Mess Solennelle," offertory, Gounod's "Ave Maria," solo by Master Hawkins. Finale, Mendelssohn’s "Wedding March." THE DRESSES OF THE LADIES. ... THE WEDDING GUESTS. ... THE WEDDING BREAKFAST Was served at the residence of the parents of the bride, 18, Carlton House-terrace. The Prince of Wales arrived there soon after the bride and bridegroom, when the Hungarian band, stationed in the corridor at the top of the grand staircase, played the "National anthem." Here, again, the floral decorations were superb. The table at which the Prince of Wales sat, with the bride and groom and their immediate relatives, was a mass of white and gold. The bridal colouring was supplied in the first instance by the colossal frosted structure of the cake, flanked by crescents of white orchids, orange flower, and branches of orange trees, bearing double burdens of fruit and bloom. The other six tables in the dining-room were decorated with white and the palest of blush pink, a favourite colour with the bride. The principal drawing-room was made to appear almost tropical with gorgeously-hued orchids and foliage, combining tints of vivid scarlet and bright yellow, with hundreds of shades of tender green. A stand of white lilies and pink roses placed in the entrance hall was much and deservedly admired. Nor were the charms of a second drawing-room, adorned with dozens upon dozens of double and single daffodils, passed over in silence. In the ball-room where the wedding presents were "on view," was an atmosphere laden with the subtle sweetness of fair white lilies. everywhere [sic] there were flowers, and the effect, combined with that of the brilliant sunshine which crowned the day, was to banish the recollection of January and suggest a vision of June. About three o’clock the Duque de Santona and his bride left by special train for Wadhurst Park, where they will pass the early days of the honeymoon. The bride went away in a costume of grey velvet, of a delicate shade, draped with ''Sicilienne'', and trimmed with rich steel passementerie; a broad band  of grey velvet edged the skirt, and she wore a jacket and hat to match, and sable boa and muff. THE WEDDING PRESENTS ...</blockquote>This report from the ''Tunbridge Wells Journal'' is a good example of a regional paper covering an event from a local perspective.<blockquote>On Wednesday morning, the Brompton Oratory was crowded to witness the nuptials of Miss Clarita de Murietta, daughter of the Marques and Marquesa de Santurce, of Wadhurst Park, Sussex, and Carlton House terrace, and the Duc de Santona. The event was invested with considerable interest to Kent and Sussex, the bride being connected by the marriage of her sister to Lord W. Nevill, with the families of Nevill, Gathorne-Hardy, Brassey, and Egerton. The Oratory was beautifully decorated with palms and ferns and white flowers. The service was fully choral and during the celebration of the nuptial mass selected music was rendered. The Prince of Wales, attended by Colonel Teesdale, was present. The Bishop or Southwark, assisted by Father Gordon and the Fathers of the Oratory, officiated. The bridge [sic] was conducted to the high altar by her father, the Marques de Santurce, who gave her away. A cousin of the bride, Mr Francis de Murietta, accompanied the Duc de Santona as best man. The bridesmaids wore dresses of soft cream crêpon, with sleeves and vests in vert pomme velvet edged with gold braid, straight skirts edged with the same, and slashes of pale mauve moiré. Two of the bridesmaids wore bonnets and the other four hats of green velvet  ornamented with gold braid and green ostrich feathers. The bridegroom's presents were a diamond arrow piercing a diamond and sapphire horseshoe, surmounted by ducal coronet in rubies and diamonds, and a "good-year" bouquet of lilies of the valley and green orchids tied with mauve streamers. The bridesmaids were Mdlle. Mara [?] Mitzans [s/b Mitjans], slater of the bridegroom, Miss Constance Barrow, daughter of Captain and Mrs Barrow, Miss Ivan Marion Nevill, daughter of Lord and Lady H. G. R. Nevill, Miss Violet Gathorne-Hardy, daughter of the Hon. C. G. and Lady Cicely Louisa Gathorne-Hardy, the Hon. Muriel Agnes Brassey, and the Hon. Marie Adelaide Brassey, daughters of Lord Brassey. The bride's dress was of rich white Venetian brocade, with long train, the front being bordered with fringes and bouquets of orange blossoms. The bodice had a high collar of lace, and a fichu of lace and fine muslin, softly draped. She wore a small crown of real orange blossoms, surmounted by a fine tulle veil. After the ceremony, the wedding party proceeded to 18, Carlton House terrace, the town residence of the Marques and Marquisa de Santurce, where a déjeuner and reception were held, the Hungarian band playing selected airs on the grand staircase. The honeymoon is to be spent at Wadhurst Park. The bride's going away dress was of a delicate shade of grey velvet, draped with Siciliéune, and trimmed with rich steel passementerie, with hat to match. The following is a list of the more important presents:— [Col. 6b–c] To the Bride. —<ref name=":1">"Marriage of Miss de Murietta." ''Tunbridge Wells Journal'' 6 February 1890, Thursday: 2 [of 8], Col. 6b–c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.com/image-viewer?issue=BL%2F0004632%2F18900206&page=2&article=036&stringtohighlight=mandeville+viscount. Same print title and p.</ref> [gifts list below set as unordered list to make it easier to read] * From the Prince and Princess of Wales, moonstone heart brooch, set with rubies and diamonds; * Princess Louise, Duchess of Fife, and the Duke of Fife, a set of silver-gilt salt cellars; * the Empress Eugénie, a large silver mirror; * the bridegroom, a magnificent sapphire and diamond necklace, a sapphire and diamond bracelet, ruby and diamond ring, pearl double-heart brooch, diamond pin, diamond and pearl brooch, gold and pearl studs, and gold-handled hunting crop; * the Marques and Marquesa de Santurce, a magnificent diamond tiara; * Marquesa de Santurce, diamond half-loop ring; * M. Mitzans, father of the bridegroom, three rows of large pearls, with diamond and emerald clasp; * Marquesa de Manzanedo, mother of bridegroom, ruby and diamond cluster necklace; * Lord and Lady William Nevill, handsome dressing bag with tortoiseshell and gold fittings, and enamel and diamond watch bracelet; * Mr Christobal and Mr Adriano de Murietta, diamond aigrette, and two pairs of carriage horses; * Mr Muriano de Murietta, silver-gilt tea service and tray, three sliver-gilt candelabras, and set of silver-gilt dessert plates; * the Duc and Duchesse de Vistahamosa, complete set of silver-gilt dessert forks, knives and spoons, and four pairs of old English silver candlesticks; * Mr Francis and Mr C. J. de Murietta, pair of large silver-gilt repoussé dishes; * Marquess de Ia Torreville, diamond double-heart brooch; * M. Calderon, set of diamond hairpins; * Marchioness of Abergavenny, black point lace and tortoiseshell fan, with monogram in rubies and diamonds; * Duke and Duchess of Newcastle, diamond and emerald watch-bracelet; * Mr Alfred de Rothschild, diamond and pearl bracelet; * Lord and Lady Sandhurst, gold "1890" sleeve links; * Mdlle. Maria Mitzans, large carbuncle and diamond pendant; * the Spanish Ambassador, six sliver coffee cups; * Duchess of Manchester, pair of silver-gilt-topped scent bottles; * Duchess of Leinster, old china box; * Hon. Hobert and Mrs Spencer, diamond pin; * Marquess of Abergavenny, writing set and candlestick; * Countess of Denbigh, embroidered silk blotting book; * Lady Alice Morland, silver vase; * Lord Richard Nevill, pigskin luncheon bag, with complete silver fittings; * Hon. Tomas and Lady Idina Brassey, leather carriage card-case with gold mountings and monogram; * Lord and Lady Brooke, Dutch silver ornament; * Viscount and Lady Violet Dangen, old painted snuff box; * Sir Oscar Clayton, pair of silver ash trays; * Sir William Gordon Cumming, silver-headed owl scent bottle; * [[Social Victorians/People/Mandeville|Viscountess Mandeville]], small green leather travelling bag; * Mrs Coupland, silver crucifix; * Canon Searle, benetier [font for holy water]; * Lady Brougham and Vaux, old gold-mounted scent bottle; * Lady Clifford and Miss Towneley, silver looking-glass; * Lord Asburton, silver box; * Lady Algernon Gordon Lennox, inlaid tortoiseshell box; * Dowager Duchess of Newcastle, silver tray; * Lord and Lady Charles Beresford, small Louis XVI. table; * Sir Frederick and Lady Milner, silver tray; * Sir Bache Cunard, scent bottle with gold top set with diamonds; * Mrs G. Cavendish-Bentick [sic], pair of small old silver vases; * Mr and Mrs  Edward Sassoon, Sèvres china inkstand; * Mr J. B. and Lady Rose Leigh, Chippendale table; * Lord and Lady Henry Nevill, complete silver tea set; * Mr Horace Farquhar, two girandoles; * Colonel Ralph Vivian, umbrella with gold handle; * the servants at Wadhurst Park and 18, Carlton House terrace, large silver salver; * the Wadhurst tenants and tradesmen, large silver cup; * Wadhurst Park stablemen, driving whip and hunting crop. To the Bridegroom. — * From the bride, turquoise and diamond pin, ruby and diamond ring; * Marquesa de Santurce, a brougham, sapphire and diamond pin, ruby and diamond horseshoe pin; * Mr C. and Mr A. de Murietta, pair of guns; * Lord and Lady William Nevill, set of crystal and sliver liqueur cups and bottles on crystal and silver tray; * Count Terro-Arias, silver-mounted paper knife; * Duc de Alba, phaeton; * Mr J. Grenfell, silver cigarette case; * Lord Richard Nevill, leather box, containing cards, counters, &c.; * Sir W. Gordon Cumming, old silver box; * Duchess de Alto, set of books; * M. and Mdme. Escandon, sapphire and diamond pin; * Count Estrados, gold match box.<ref name=":1" /> </blockquote> == Notes and Questions == # ==References== {{reflist}} 1lyzjpgc18ksxmvwp6t08pfk8d5l341 0 332308 2835524 2835422 2026-10-02T16:25:27Z Howie2024 2995240 Distinguishing between existing time-traveler UFO hypothesis. 2835524 wikitext text/x-wiki {{Research project}} = Time-Loop UFO/UAP Hypothesis = '''Version 1.1 — October 2026''' == Overview == The '''Time-Loop UFO/UAP Hypothesis''' is a speculative interdisciplinary research framework examining whether an apparently anomalous observation could, in a hypothetical scenario, become part of a self-consistent loop involving information, artificial intelligence (AI), augmented reality (AR), human action, and historical records. The hypothesis does '''not''' propose that unidentified anomalous phenomena (UAP) are extraterrestrial spacecraft, time travelers, future spacecraft, or evidence of backward time travel. Instead, UAP are treated here as an '''observational category''': observations that have not yet been satisfactorily identified. The central question of this article is therefore narrower: '''If future-origin information were ever independently demonstrated, could it be modeled as part of a self-consistent temporal information loop?''' This is a hypothetical question, not an established scientific claim. The framework builds a conceptual bridge between: * UAP observation; * sensor data; * information processing; * artificial intelligence; * augmented reality; * human interpretation and action; * historical information; * causal feedback; * closed timelike curves; * the Novikov self-consistency principle; and * the broader research framework developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The article distinguishes among three levels of discussion: * '''Established science''' — established methods and findings concerning observation, information systems, artificial intelligence, augmented reality, causality, and theoretical physics. * '''Theoretical models''' — mathematical and physical models involving closed timelike curves, self-consistency, retrocausality, and information. * '''Speculative hypotheses''' — proposed extensions concerning future-origin information, UAP, AI, AR, and temporal information loops. The speculative portions of this article should not be interpreted as evidence that such phenomena exist. == Relationship to the Main Research Framework == This article is a supporting component of the broader research project developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The main article develops a general framework connecting: * temporal consistency; * causality; * information integrity; * artificial intelligence; * augmented reality; and * ethical constraints. The present article develops a more specific observational branch of that framework. The conceptual progression can be represented as: <math> \text{physical event} \rightarrow \text{sensor observation} \rightarrow \text{recorded data} \rightarrow \text{AI processing} \rightarrow \text{interpretation} \rightarrow \text{human action} \rightarrow \text{historical information} </math> The speculative temporal extension is: <math> I_{\mathrm{future}} \rightarrow AR_{\mathrm{past}} \rightarrow A_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> This article therefore asks whether a hypothetical temporal information loop could be studied using the same concepts of physical consistency, informational consistency, provenance, causal minimalism, and ethical consistency developed in the main research framework. == Background == === UAP as an Observational Category === The term '''unidentified anomalous phenomena''' refers to observations that remain unidentified after available information and analysis have been considered. The term does not itself specify a cause. A UAP observation may potentially result from: * an aircraft or other conventional human technology; * atmospheric phenomena; * astronomical objects; * sensor artifacts; * optical effects; * viewing geometry; * motion of the observing platform; * image or video-processing effects; * incomplete or inaccurate metadata; * unfamiliar but conventional technology; * limitations in observation or measurement; or * phenomena that remain unexplained after available evidence has been examined. Consequently: <math> \text{unidentified observation} \neq \text{identified cause} </math> and: <math> \text{unexplained} \neq \text{evidence for time travel} </math> This distinction is fundamental to the present hypothesis. === Causality and Temporal Consistency === Ordinary physical reasoning assumes that causes precede their effects within the relevant causal structure. Relativity distinguishes timelike, null, and spacelike relationships between events. General relativity admits mathematical spacetime solutions containing closed timelike curves under certain conditions. A closed timelike curve is a worldline that returns to an earlier event in the spacetime description. The existence of such mathematical solutions does not establish that usable time travel occurs in nature. Research concerning closed timelike curves has nevertheless raised an important theoretical question: '''If a causal loop existed, what restrictions would be required for the complete history to remain physically consistent?''' The Novikov self-consistency principle is one proposed answer. In its basic form, a local physical solution involving a closed timelike curve must be extendable into a globally self-consistent solution. This provides a theoretical foundation for examining hypothetical information loops without assuming that arbitrary changes to the past are possible. === Information and Self-Consistency === A temporal paradox can be considered not only in terms of physical objects but also in terms of information. Suppose a hypothetical system received information from a future state and that information influenced events in an earlier state. If those earlier events subsequently contributed to the future state that generated the original information, a feedback loop could arise: <math> I_{\mathrm{future}} \rightarrow S_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> The important question would then be whether the complete information history is self-consistent. This does not establish that such a loop can occur. It provides a mathematical and conceptual structure in which such a possibility could be examined. == From Observation to Interpretation == A central methodological distinction in this research is the difference between an observation and an interpretation. A simplified observation pipeline is: <math> O \rightarrow D \rightarrow P \rightarrow I \rightarrow H </math> where: * <math>O</math> = physical occurrence; * <math>D</math> = sensor data; * <math>P</math> = processing; * <math>I</math> = interpretation; and * <math>H</math> = human understanding or hypothesis. Each stage can introduce uncertainty. For example, a camera does not directly record an object's complete physical state. It records signals produced by an imaging system under particular exposure, optical, geometric, atmospheric, and instrumental conditions. Processing can then modify the representation. AI can classify, enhance, reconstruct, interpolate, summarize, or otherwise transform the information. Human observers subsequently interpret the resulting representation. Therefore: <math> \text{observation} \neq \text{description} \neq \text{processing} \neq \text{interpretation} \neq \text{hypothesis} </math> This distinction becomes especially important when AI-generated or AI-enhanced imagery is involved. An apparently unusual image may represent: * a physical phenomenon; * a sensor artifact; * a processing artifact; * an AI-generated reconstruction; * an AI classification; * a combination of physical and synthetic information; or * an interpretation imposed upon incomplete data. A scientifically useful investigation should preserve these distinctions. == Artificial Intelligence as an Information Layer == Modern information systems can transform observations into increasingly complex representations. A simplified AI pipeline may be expressed as: <math> D_{\mathrm{sensor}} \rightarrow M_{\mathrm{AI}} \rightarrow R_{\mathrm{AI}} \rightarrow H_{\mathrm{human}} </math> where: * <math>D_{\mathrm{sensor}}</math> is sensor data; * <math>M_{\mathrm{AI}}</math> is an AI model or processing system; * <math>R_{\mathrm{AI}}</math> is the resulting representation, classification, prediction, or reconstruction; and * <math>H_{\mathrm{human}}</math> represents human interpretation. AI therefore creates an additional information layer between physical measurement and human understanding. The distinction between raw measurement and processed representation becomes increasingly important when an observation is used to support a claim about an unusual physical event. This produces an important research question: '''How can an observer distinguish information contained in an observation from information introduced by the system interpreting the observation?''' This question does not require time travel. It is already relevant to contemporary AI-assisted image and video analysis. == Augmented Reality and the Information Layer == Augmented reality combines information generated by a digital system with a user's perception of a physical environment. A simplified AR information pathway can be written as: <math> W \rightarrow S \rightarrow D \rightarrow A \rightarrow P \rightarrow H </math> where: * <math>W</math> = physical world; * <math>S</math> = sensors; * <math>D</math> = digital information; * <math>A</math> = augmented-reality overlay; * <math>P</math> = human perception; and * <math>H</math> = human action or interpretation. AR therefore introduces an important distinction between: '''what is physically present''' and: '''what the information system tells the observer about what is physically present'''. The presence of an AR layer would therefore make interpretation more complex, not less. == The AI and AR Feedback System == AI and AR can be considered together as an information-processing system: <math> \text{physical world} \rightarrow \text{sensors} \rightarrow \text{AI processing} \rightarrow \text{digital information} \rightarrow \text{AR display} \rightarrow \text{human perception} \rightarrow \text{human action} </math> Human action can then produce new physical observations. This creates an ordinary feedback structure: <math> \text{information} \rightarrow \text{human action} \rightarrow \text{new observation} \rightarrow \text{new information} </math> Nothing about this ordinary feedback loop requires time travel. It is already a characteristic of many information systems. The significance for the present hypothesis is that the same structure provides a conceptual pathway through which a hypothetical temporal information loop could be represented. == The Time-Loop UAP Hypothesis == The hypothesis can be stated conservatively: '''If a future information-processing system could somehow provide information to an earlier historical period, and if that information influenced observations or actions that later contributed to the future system's own informational state, the resulting process could be modeled as a temporal information loop.''' The hypothesis does not require that the information arrive through a physical spacecraft. It could, in principle, be represented as: * an image; * a measurement; * a prediction; * a coordinate; * a symbolic message; * an AI-generated representation; * an AR overlay; or * another information-bearing signal. The crucial hypothetical property would be its '''temporal provenance'''. The question would not simply be: '''Was a UAP observed?''' Instead, it would be: '''Can the information associated with the observation be demonstrated to have originated from a future state of the same causal system?''' That is a much stronger requirement. No presently established UAP evidence demonstrates this. == Relationship to the Existing Time-Traveler UFO Hypothesis == The present research is related to the broader '''time-traveler UFO hypothesis''' but does not simply propose that UAP are future humans. Instead, it investigates whether a hypothetical '''future-origin information process''' could be represented as a self-consistent temporal information loop involving artificial intelligence (AI), augmented reality (AR), human action, and historical information. The distinction can be expressed as follows: <math> \text{time-traveler UFO hypothesis} \rightarrow \text{UFO/UAP as future human technology} </math> The present research considers a different and more general possibility: <math> \text{future-origin information} \rightarrow \text{past information system} \rightarrow \text{human action} \rightarrow \text{historical information} \rightarrow \text{future information} </math> The present framework therefore does not require that a UAP be a vehicle carrying people from the future. A hypothetical temporal effect could instead involve information, measurement, prediction, artificial intelligence, augmented reality, or another information-processing system. This distinction is important because an '''unidentified observation''' does not by itself establish either extraterrestrial origin or future-human origin. The purpose of the present research is to examine whether the broader concept of temporal information feedback can be formulated mathematically and analyzed using concepts such as causality, closed timelike curves, self-consistency, information provenance, and constrained probability states. The existing time-traveler UFO hypothesis provides a related conceptual background, while the present research develops a more specific information-centered framework. No claim is made that UAP currently provide evidence for future-origin information or temporal information loops. == Temporal Information Feedback == The central proposed feedback structure is: <math> I_{\mathrm{future}} \rightarrow AR_{\mathrm{past}} \rightarrow A_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> where: * <math>I_{\mathrm{future}}</math> = hypothetical future-origin information; * <math>AR_{\mathrm{past}}</math> = an augmented-reality or information display in the past; * <math>A_{\mathrm{past}}</math> = human action influenced by that information; and * <math>H_{\mathrm{history}}</math> = the resulting historical record. The resulting information could then become part of the historical data available to the future system. This would constitute a hypothetical '''temporal information feedback loop'''. No such loop has been established experimentally. == From Temporal Information to a UAP Hypothesis == The UAP component of the hypothesis asks whether an apparently anomalous observation could, under a hypothetical temporal-information model, be associated with such a loop. The question is not: '''Are UAPs time travelers?''' Instead, the more limited question is: '''Could an apparently anomalous observation, if future evidence eventually demonstrated that it contained information or technology originating from a future period, be modeled as part of a self-consistent temporal information loop?''' This formulation deliberately postpones the question of origin. An observation could initially remain unexplained while alternative hypotheses are investigated, including: # known aircraft or other human technology; # atmospheric or astronomical phenomena; # sensor artifacts; # viewing geometry; # errors in measurement or interpretation; # image or video processing effects; # incomplete metadata; # conventional technological explanations not initially recognized; and # genuinely unexplained phenomena. A temporal explanation would therefore require substantially stronger evidence than simply observing something unusual. == Physical, Informational, and Interpretive Consistency == The broader research framework distinguishes several types of consistency. === Physical consistency === A hypothetical temporal process would have to satisfy whatever physical laws govern the relevant spacetime. If closed timelike curves were involved, this could include global self-consistency constraints. === Informational consistency === The information appearing at different points in the loop would have to remain mutually compatible. If a future system received a historical record containing information that originated from that future system, the complete information history would need to be examined for consistency. === Interpretive consistency === The interpretation of an observation must remain distinguishable from the underlying measurement. An AI or AR system could alter what an observer perceives without altering the underlying physical event. Therefore: <math> \text{physical event} \neq \text{sensor representation} \neq \text{AI representation} \neq \text{human interpretation} </math> === Ethical consistency === If a future information system could influence historical human decisions, ethical constraints would become relevant. A hypothetical system with access to causally sensitive historical information would raise questions concerning: * information integrity; * unnecessary intervention; * consent; * manipulation; * historical authenticity; * unintended consequences; * accountability; and * preservation of human agency. This connects directly to the broader framework developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. == A Proposed Framework == The following principles are proposed as a conceptual framework. They are not established physical laws. === 1. Information integrity === Measurements, metadata, processing steps, and interpretations should be preserved separately. A future investigation should ideally retain: * original sensor files; * calibration information; * timestamps; * sensor location; * sensor orientation; * exposure information; * instrument characteristics; * processing history; * AI-model information; * transformation records; and * human annotations. === 2. Causal minimalism === If a system had access to causally sensitive information, it should minimize unnecessary intervention. Conceptually: <math> \text{unnecessary intervention} \rightarrow \text{additional causal pathways} \rightarrow \text{greater feedback complexity} </math> In a hypothetical temporal system, minimizing unnecessary intervention could reduce opportunities for unstable or contradictory information loops. === 3. Provenance preservation === Every significant information object should have a traceable origin. For a UAP investigation, this means distinguishing: <math> P_{\mathrm{physical}} \rightarrow P_{\mathrm{sensor}} \rightarrow P_{\mathrm{processing}} \rightarrow P_{\mathrm{AI}} \rightarrow P_{\mathrm{interpretation}} </math> The purpose is to determine which portions of the final representation can be traced to physical measurement and which are products of later processing. === 4. Predictive transparency === A system should distinguish between information that describes an event and information that may influence the event. This distinction becomes important when predictions are shown to people through AI systems or AR displays. The ordinary feedback structure is: <math> \text{prediction} \rightarrow \text{behavior} \rightarrow \text{new data} \rightarrow \text{prediction} </math> A hypothetical temporal system would extend this into: <math> \text{prediction} \rightarrow \text{future action} \rightarrow \text{historical intervention} \rightarrow \text{changed data} \rightarrow \text{prediction} </math> === 5. Self-consistency === If a temporal information loop were ever demonstrated, the complete information history would need to be evaluated rather than examining isolated events. Conceptually: <math> H_{\mathrm{complete}} \in C_{\mathrm{physical}} \cap C_{\mathrm{informational}} \cap C_{\mathrm{interpretive}} \cap C_{\mathrm{ethical}} </math> where the four sets represent different admissibility conditions. This extends the idea of self-consistency into an interdisciplinary framework. == A Possible Temporal AR Loop == A particularly interesting thought experiment involves AR. Imagine that a future information-processing system possesses highly detailed historical information. Suppose, hypothetically, that the system somehow provides a representation of an event to an observer in an earlier period. The observer sees the information through an AR system. The observer then acts on that information. The action becomes part of the historical record. The future system subsequently learns from that historical record. The conceptual loop is: <math> I_F \rightarrow AR_P \rightarrow A_P \rightarrow H \rightarrow AI_F \rightarrow I_F </math> where: * <math>I_F</math> = future information; * <math>AR_P</math> = past AR representation; * <math>A_P</math> = past action; * <math>H</math> = historical record; and * <math>AI_F</math> = future AI system. This is a thought experiment. It does not establish that such a mechanism exists. == Bootstrap Information and the Origin Problem == A classic information-loop problem arises when information exists in a causal loop without an identifiable external origin. For example: <math> I_1 \rightarrow I_2 \rightarrow I_3 \rightarrow I_1 </math> If the information contained in <math>I_1</math> is completely determined by later information in the same loop, it may become difficult to identify an independent source for the information. This does not automatically constitute a contradiction. However, it creates a major epistemological problem. A scientific investigation would need to distinguish: * information genuinely originating from an external source; * information generated internally by the system; * information reconstructed from incomplete observations; * information introduced by human interpretation; * information produced by AI systems; and * information that only appears to have a future origin because its provenance is incomplete. The bootstrap problem therefore makes provenance especially important. == UAP Observation Versus UAP Interpretation == The framework proposes maintaining separate levels of description. '''Level 1 — Observation''' What was directly measured or recorded? '''Level 2 — Description''' What does the recording objectively show? '''Level 3 — Processing''' What transformations, enhancement, classification, or reconstruction were applied? '''Level 4 — Interpretation''' What explanations are consistent with the available evidence? '''Level 5 — Hypothesis''' What broader models could potentially explain the observation? A temporal-loop explanation belongs at Level 5. It should not be inserted into Level 1 or Level 2 merely because an observation appears unusual. == Relationship to Augmented Reality == AR introduces an additional methodological problem because the displayed scene can contain both physical and digital information. Conceptually: <math> R_{\mathrm{display}} = R_{\mathrm{physical}} + I_{\mathrm{digital}} </math> where: * <math>R_{\mathrm{display}}</math> represents the displayed perceptual environment; * <math>R_{\mathrm{physical}}</math> represents information derived from the physical environment; and * <math>I_{\mathrm{digital}}</math> represents digitally generated or retrieved information. This equation is a conceptual representation rather than a physical law. If the digital information were generated by an AI system, the observer could potentially perceive an object, annotation, prediction, reconstruction, or other representation that is not identical to a directly observed physical object. Therefore, an AR-mediated observation would require careful preservation of the underlying sensor data and metadata before conclusions were drawn about the physical phenomenon. == Information Provenance and Historical Integrity == The hypothesis requires unusually strong attention to provenance. For an information object <math>I</math>, provenance can be represented conceptually as: <math> \Pi(I) = \{t,s,m,p,a,r\} </math> where: * <math>t</math> = timestamp; * <math>s</math> = source; * <math>m</math> = metadata; * <math>p</math> = processing history; * <math>a</math> = analytical actions; and * <math>r</math> = resulting representation. A temporal-origin claim would require showing that the provenance chain cannot be explained by ordinary forward information flow. This is a much stronger requirement than demonstrating that a prediction happened to be correct. == What Would Count as Evidence? == The Time-Loop UAP Hypothesis should require substantially stronger evidence than an unusual image or unexplained observation. Potentially relevant evidence would include: === 1. Independent physical measurement === The phenomenon should be recorded by calibrated instruments capable of producing quantitative measurements. === 2. Multiple synchronized observations === Independent sensors should observe the same event with synchronized timing. === 3. Complete metadata === The record should include sufficient information to reconstruct the observational circumstances. === 4. Preservation of original data === Original sensor data should be preserved rather than relying exclusively on screenshots, compressed video, or later reconstructions. === 5. Independent analysis === The observation should be analyzable by researchers who are not dependent upon the original interpretation. === 6. Demonstrable information provenance === If future-origin information is proposed, the information should have a demonstrable provenance that cannot be explained by ordinary information transfer. === 7. Temporal priority === The allegedly future-origin information would need to be demonstrably present before the event or information from which it would ordinarily have been derived. This is crucial. A prediction generated after an event cannot ordinarily establish that the information originated from the future. === 8. Reproducibility === The effect should be observable or testable independently. === 9. Falsifiable predictions === The hypothesis should produce predictions that distinguish it from competing explanations. === 10. Exclusion of conventional explanations === Sensor artifacts, atmospheric effects, astronomical objects, viewing geometry, image processing, human technology, and other ordinary explanations should be investigated before a temporal explanation is considered. == Alternative Explanations and Falsifiability == The hypothesis should remain subordinate to ordinary explanations until evidence requires otherwise. Potential explanations include: * conventional aircraft or technology; * atmospheric or astronomical phenomena; * sensor artifacts; * image-processing effects; * viewing-geometry effects; * AI interpretation; * AR-generated information; * incomplete metadata; * conventional informational feedback; and * genuinely unexplained phenomena. A temporal explanation would require evidence that distinguishes it from these alternatives. An unresolved observation therefore remains an unresolved observation until additional evidence establishes a specific explanation. == Mathematical Formulation == The equations in this section are proposed mathematical representations of the conceptual framework. They are not equations derived from quantum mechanics or established models of time travel. === State-transition representation === Let <math>S_t</math> represent the information state of a system at time <math>t</math>. An ordinary forward process can be represented as: <math> S_{t+1}=F_t(S_t,A_t) </math> where <math>A_t</math> represents actions or interventions at time <math>t</math>. A hypothetical temporally coupled system could be represented as: <math> S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t) </math> where <math>B_t</math> represents additional boundary or environmental information. The future-state dependence is the speculative element. === Trajectory and fixed-point representation === A complete trajectory may be written: <math> S=(S_0,S_1,\ldots,S_T) </math> A self-consistent trajectory can be represented by: <math> S^{*}=\Phi(S^{*}) </math> This means that the complete history reproduces itself under the transformation and constraints represented by <math>\Phi</math>. This is a mathematical analogy rather than an established physical law. === Physical, informational, interpretive, and ethical admissibility === The proposed framework can be represented as: <math> S^{*} \in C_{\mathrm{physical}} \cap C_{\mathrm{informational}} \cap C_{\mathrm{interpretive}} \cap C_{\mathrm{ethical}} </math> A history would therefore be considered admissible in the model only if it satisfies all four classes of constraint. == Probability-Based State Transformations == A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems. Let: <math> p_t(x)\geq0 </math> and: <math> \sum_x p_t(x)=1 </math> where <math>p_t(x)</math> represents the probability of information state <math>x</math> at time <math>t</math>. A probabilistic transformation can be written as: <math> \widetilde{p}_{t+1}(x) = T_t(p_t,x) </math> If the transformed distribution is not normalized, define: <math> Z_{t+1} = \sum_x \widetilde{p}_{t+1}(x) </math> and: <math> p_{t+1}(x) = \frac{\widetilde{p}_{t+1}(x)} {Z_{t+1}} </math> This provides an abstract analogy: '''transformation → constraint → admissible state''' However, probability normalization alone is not equivalent to temporal consistency. Normalization ensures that a probability distribution sums to one. It does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible. A trajectory-level admissibility condition can be represented as: <math> \mathcal{A}_{\mathrm{total}} = \mathcal{A}_{\mathrm{phys}} \cap \mathcal{A}_{\mathrm{info}} \cap \mathcal{A}_{\mathrm{interp}} \cap \mathcal{A}_{\mathrm{eth}} </math> where: * <math>\mathcal{A}_{\mathrm{phys}}</math> represents physically admissible trajectories; * <math>\mathcal{A}_{\mathrm{info}}</math> represents informationally consistent trajectories; * <math>\mathcal{A}_{\mathrm{interp}}</math> represents interpretation-consistent trajectories; and * <math>\mathcal{A}_{\mathrm{eth}}</math> represents ethically admissible trajectories. A constrained probability distribution could then be written conceptually as: <math> P_{\mathrm{adm}}(x) = P(x\mid x\in\mathcal{A}_{\mathrm{total}}) </math> This remains an abstract modeling analogy. It does not imply that ethical rules are physical laws or that quantum mechanics requires the proposed ethical constraints. == Relationship to Closed Timelike Curves == The hypothesis draws conceptual inspiration from research concerning closed timelike curves. In a spacetime containing a closed timelike curve, a worldline can in principle return to an earlier spacetime event. The important theoretical issue is whether a complete solution can remain self-consistent. Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever examined the Cauchy problem in spacetimes containing closed timelike curves and discussed the need for a global self-consistency condition if such curves are permitted. Echeverria, Klinkhammer, and Thorne subsequently examined a classical billiard-ball model involving wormhole-generated closed timelike curves and found examples involving multiple self-consistent solutions. The Time-Loop UAP Hypothesis does not assert that UAP observations involve closed timelike curves. Instead, closed-timelike-curve research supplies a theoretical vocabulary for asking how a causal loop might remain globally consistent. == Relationship to the Novikov Self-Consistency Principle == The [[Novikov Self-Consistency Principle]] provides an especially relevant conceptual connection. The principle can be summarized as: <math> \text{local possibility} \rightarrow \text{global consistency requirement} \rightarrow \text{admissible history} </math> Applied only as a speculative analogy, the principle suggests that if a temporal information loop existed, not every apparently possible information sequence would necessarily correspond to an admissible history. The hypothesis therefore does not assume: '''The past can be changed arbitrarily.''' Instead it asks: '''If future-origin information existed, what global constraints would be required for the complete information history to remain self-consistent?''' This is a substantially narrower proposition. == Relationship to Temporal Consistency and Quantum Ethics == The strongest conceptual relationship is with [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The main article identifies a progression: <math> \text{physical consistency} \rightarrow \text{informational consistency} \rightarrow \text{ethical consistency} </math> The present UAP hypothesis adds another layer: <math> \text{physical observation} \rightarrow \text{information} \rightarrow \text{interpretation} \rightarrow \text{action} \rightarrow \text{historical information} </math> The two frameworks can therefore be combined conceptually. A hypothetical future AI system might not need to physically transport an object through time. It would only need, in the speculative model, to affect information that later becomes part of its own historical input. This produces the possibility of: <math> \text{information} \rightarrow \text{action} \rightarrow \text{history} \rightarrow \text{future information} </math> The main article already considers a related structure involving augmented reality: '''AR information → human behavior → historical record → future training data''' The present article extends that conceptual structure into the UAP domain. The important distinction remains: '''ordinary informational feedback is already real; a temporal feedback loop remains hypothetical.''' == Ethical Implications of a Hypothetical Temporal Information System == If a future information-processing system could affect historical information, the ethical problem would extend beyond physics. A hypothetical system might face questions such as: * Should it intervene at all? * How much intervention would be justified? * Should it disclose the origin of information? * Could historical people give meaningful consent? * What happens if the information changes subsequent decisions? * Who is responsible for unintended consequences? * How should historical records be protected from deliberate manipulation? * Should an AI system be permitted to optimize outcomes by altering information conditions? These questions are consistent with the broader concept of '''ethical consistency''' developed in the main research article. The ethical framework does not depend on proving time travel. Similar questions already arise when AI systems influence human behavior and subsequently learn from the resulting behavior. == Science Fiction and Conceptual Analogies == Science fiction can provide useful conceptual analogies, but it should not be treated as scientific evidence. The broader research project uses [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]] to examine fictional representations of: * advanced AI; * information asymmetry; * autonomous decision-making; * objectives and constraints; * human-machine interaction; and * consequences of information systems. The present article does not claim that fictional depictions of UFOs or time travel predict physical reality. Fiction can instead serve as a conceptual laboratory in which questions concerning causality, information, AI, and temporal consistency can be examined. == Research Questions == The following questions remain open within the speculative framework: # Can UAP investigation methods be designed to preserve complete sensor and processing provenance? # How can AI-generated interpretations be distinguished from direct sensor measurements? # How should AR systems distinguish physical observations from digital overlays? # Can information-feedback loops be modeled using established causal and information-theoretic mathematics? # Under what assumptions can a temporal information loop be represented as a fixed-point problem? # What conditions would be required for a future-origin information claim to be empirically testable? # Could a future information system create a logically self-consistent information loop without producing an observable contradiction? # Would such a loop necessarily involve a physical closed timelike curve, or could a different theoretical mechanism produce an equivalent information structure? # How should bootstrap information be treated within a causal model? # What would distinguish a genuine temporal information effect from an ordinary prediction, coincidence, data leak, or reconstruction? # Could probability-based state transformations provide a useful mathematical language for studying constrained temporal information systems? # What ethical constraints would be appropriate for an information-processing system capable of influencing historical conditions? == Limitations and Open Questions == This article has substantial limitations. First, no established experiment demonstrates controllable communication from the future into the past. Second, no established evidence demonstrates that UAP are caused by time travel. Third, unexplained observations do not by themselves establish a new physical mechanism. Fourth, AI and AR can introduce additional layers of interpretation between a physical event and human perception. Fifth, the mathematical formulations in this article are proposed models and analogies rather than experimentally validated physical equations. Sixth, a self-consistent mathematical model does not demonstrate that the modeled process occurs in nature. Seventh, a correct prediction does not by itself establish future-origin information. Finally, the hypothesis may ultimately prove unfalsifiable or physically impossible if no mechanism exists by which future-origin information can influence past observations. These limitations are not weaknesses to be hidden; they are part of the scientific definition of the hypothesis. == Conclusion == The Time-Loop UAP Hypothesis proposes a conservative question rather than a conclusion. UAP are treated as observations rather than explanations. The research begins with a standard information pathway: <math> \text{physical event} \rightarrow \text{sensor} \rightarrow \text{data} \rightarrow \text{processing} \rightarrow \text{interpretation} </math> AI and AR introduce additional information-processing layers. An ordinary feedback system can then arise: <math> \text{prediction} \rightarrow \text{human action} \rightarrow \text{new data} \rightarrow \text{new prediction} </math> The speculative extension is: <math> \text{future-origin information} \rightarrow \text{past information system} \rightarrow \text{human action} \rightarrow \text{historical record} \rightarrow \text{future information} </math> If such a process were ever demonstrated, the concepts of physical consistency, informational consistency, interpretive integrity, provenance, and ethical consistency could provide a framework for analyzing it. The hypothesis therefore does not begin with the proposition that UAP are time travelers. It begins with a more general scientific question: '''Could future-origin information, if independently demonstrated, form part of a globally self-consistent information loop?''' At present, this remains a speculative research question. == References == === Physics and temporal consistency === * Friedman, J. L., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., and Yurtsever, U. (1990). "Cauchy problem in spacetimes with closed timelike curves." ''Physical Review D'', 42, 1915–1930. [https://doi.org/10.1103/PhysRevD.42.1915 DOI] * Echeverria, F., Klinkhammer, G., and Thorne, K. S. (1991). "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory." ''Physical Review D'', 44, 1077–1099. [https://doi.org/10.1103/PhysRevD.44.1077 DOI] * Hawking, S. W. (1992). "Chronology protection conjecture." ''Physical Review D'', 46, 603–611. [https://doi.org/10.1103/PhysRevD.46.603 DOI] * Deutsch, D. (1991). "Quantum mechanics near closed timelike lines." ''Physical Review D'', 44, 3197–3217. [https://doi.org/10.1103/PhysRevD.44.3197 DOI] === UAP methodology === * NASA. (2023). ''NASA Unidentified Anomalous Phenomena Independent Study Team Final Report''. [https://science.nasa.gov/uap/ NASA UAP Independent Study] === Artificial intelligence and information systems === * National Institute of Standards and Technology. (2023). ''Artificial Intelligence Risk Management Framework (AI RMF 1.0)''. [https://www.nist.gov/itl/ai-risk-management-framework NIST AI Risk Management Framework] * National Institute of Standards and Technology. ''AI Measurement and Evaluation''. [https://www.nist.gov/ai-measurement-and-evaluation NIST AI Measurement and Evaluation] === Augmented reality === * Azuma, R. T. (1997). "A Survey of Augmented Reality." ''Presence: Teleoperators and Virtual Environments'', 6(4), 355–385. [https://doi.org/10.1162/pres.1997.6.4.355 DOI] === Fiction and artificial intelligence === * Kubrick, S., director. (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer. == Further Reading == * [[Closed Timelike Curves]] * [[Novikov Self-Consistency Principle]] * [[Causality]] * [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]] * [https://science.nasa.gov/uap/ NASA UAP Independent Study] * [https://www.nist.gov/itl/ai-risk-management-framework NIST AI Risk Management Framework] * [https://doi.org/10.1162/pres.1997.6.4.355 Azuma, "A Survey of Augmented Reality"] == Research Status == '''Research status: Speculative interdisciplinary framework.''' Version 1.1 develops a conceptual hypothesis connecting UAP observation, information provenance, artificial intelligence, augmented reality, temporal information feedback, causal consistency, and ethical constraints. The hypothesis does not claim that UAP are extraterrestrial spacecraft, time travelers, or evidence of backward time travel. No established scientific experiment currently demonstrates controllable information transfer from the future to the past. The mathematical formulations presented in this article are proposed models and analogies for further investigation. Future versions may refine the mathematical framework, develop computational simulations, examine information-theoretic constraints, and investigate whether the proposed temporal-information structures can produce experimentally distinguishable predictions. == Author and Attribution == This research article is part of an ongoing independent research project developed by Howard Richardson. The ideas presented here are exploratory and are intended to be developed, criticized, tested, refined, or rejected through further research. The publication of a speculative hypothesis on Wikiversity does not constitute scientific confirmation. The article is released for educational and research discussion under the applicable Wikiversity and Wikimedia licensing framework. 26yrvu9e129soptmjpv8jk6ok4ezfcw 2835526 2835524 2026-10-02T16:44:20Z Howie2024 2995240 /* Probability-Based State Transformations */ adding Relationship to Probability Dilation Theory 2835526 wikitext text/x-wiki {{Research project}} = Time-Loop UFO/UAP Hypothesis = '''Version 1.1 — October 2026''' == Overview == The '''Time-Loop UFO/UAP Hypothesis''' is a speculative interdisciplinary research framework examining whether an apparently anomalous observation could, in a hypothetical scenario, become part of a self-consistent loop involving information, artificial intelligence (AI), augmented reality (AR), human action, and historical records. The hypothesis does '''not''' propose that unidentified anomalous phenomena (UAP) are extraterrestrial spacecraft, time travelers, future spacecraft, or evidence of backward time travel. Instead, UAP are treated here as an '''observational category''': observations that have not yet been satisfactorily identified. The central question of this article is therefore narrower: '''If future-origin information were ever independently demonstrated, could it be modeled as part of a self-consistent temporal information loop?''' This is a hypothetical question, not an established scientific claim. The framework builds a conceptual bridge between: * UAP observation; * sensor data; * information processing; * artificial intelligence; * augmented reality; * human interpretation and action; * historical information; * causal feedback; * closed timelike curves; * the Novikov self-consistency principle; and * the broader research framework developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The article distinguishes among three levels of discussion: * '''Established science''' — established methods and findings concerning observation, information systems, artificial intelligence, augmented reality, causality, and theoretical physics. * '''Theoretical models''' — mathematical and physical models involving closed timelike curves, self-consistency, retrocausality, and information. * '''Speculative hypotheses''' — proposed extensions concerning future-origin information, UAP, AI, AR, and temporal information loops. The speculative portions of this article should not be interpreted as evidence that such phenomena exist. == Relationship to the Main Research Framework == This article is a supporting component of the broader research project developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The main article develops a general framework connecting: * temporal consistency; * causality; * information integrity; * artificial intelligence; * augmented reality; and * ethical constraints. The present article develops a more specific observational branch of that framework. The conceptual progression can be represented as: <math> \text{physical event} \rightarrow \text{sensor observation} \rightarrow \text{recorded data} \rightarrow \text{AI processing} \rightarrow \text{interpretation} \rightarrow \text{human action} \rightarrow \text{historical information} </math> The speculative temporal extension is: <math> I_{\mathrm{future}} \rightarrow AR_{\mathrm{past}} \rightarrow A_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> This article therefore asks whether a hypothetical temporal information loop could be studied using the same concepts of physical consistency, informational consistency, provenance, causal minimalism, and ethical consistency developed in the main research framework. == Background == === UAP as an Observational Category === The term '''unidentified anomalous phenomena''' refers to observations that remain unidentified after available information and analysis have been considered. The term does not itself specify a cause. A UAP observation may potentially result from: * an aircraft or other conventional human technology; * atmospheric phenomena; * astronomical objects; * sensor artifacts; * optical effects; * viewing geometry; * motion of the observing platform; * image or video-processing effects; * incomplete or inaccurate metadata; * unfamiliar but conventional technology; * limitations in observation or measurement; or * phenomena that remain unexplained after available evidence has been examined. Consequently: <math> \text{unidentified observation} \neq \text{identified cause} </math> and: <math> \text{unexplained} \neq \text{evidence for time travel} </math> This distinction is fundamental to the present hypothesis. === Causality and Temporal Consistency === Ordinary physical reasoning assumes that causes precede their effects within the relevant causal structure. Relativity distinguishes timelike, null, and spacelike relationships between events. General relativity admits mathematical spacetime solutions containing closed timelike curves under certain conditions. A closed timelike curve is a worldline that returns to an earlier event in the spacetime description. The existence of such mathematical solutions does not establish that usable time travel occurs in nature. Research concerning closed timelike curves has nevertheless raised an important theoretical question: '''If a causal loop existed, what restrictions would be required for the complete history to remain physically consistent?''' The Novikov self-consistency principle is one proposed answer. In its basic form, a local physical solution involving a closed timelike curve must be extendable into a globally self-consistent solution. This provides a theoretical foundation for examining hypothetical information loops without assuming that arbitrary changes to the past are possible. === Information and Self-Consistency === A temporal paradox can be considered not only in terms of physical objects but also in terms of information. Suppose a hypothetical system received information from a future state and that information influenced events in an earlier state. If those earlier events subsequently contributed to the future state that generated the original information, a feedback loop could arise: <math> I_{\mathrm{future}} \rightarrow S_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> The important question would then be whether the complete information history is self-consistent. This does not establish that such a loop can occur. It provides a mathematical and conceptual structure in which such a possibility could be examined. == From Observation to Interpretation == A central methodological distinction in this research is the difference between an observation and an interpretation. A simplified observation pipeline is: <math> O \rightarrow D \rightarrow P \rightarrow I \rightarrow H </math> where: * <math>O</math> = physical occurrence; * <math>D</math> = sensor data; * <math>P</math> = processing; * <math>I</math> = interpretation; and * <math>H</math> = human understanding or hypothesis. Each stage can introduce uncertainty. For example, a camera does not directly record an object's complete physical state. It records signals produced by an imaging system under particular exposure, optical, geometric, atmospheric, and instrumental conditions. Processing can then modify the representation. AI can classify, enhance, reconstruct, interpolate, summarize, or otherwise transform the information. Human observers subsequently interpret the resulting representation. Therefore: <math> \text{observation} \neq \text{description} \neq \text{processing} \neq \text{interpretation} \neq \text{hypothesis} </math> This distinction becomes especially important when AI-generated or AI-enhanced imagery is involved. An apparently unusual image may represent: * a physical phenomenon; * a sensor artifact; * a processing artifact; * an AI-generated reconstruction; * an AI classification; * a combination of physical and synthetic information; or * an interpretation imposed upon incomplete data. A scientifically useful investigation should preserve these distinctions. == Artificial Intelligence as an Information Layer == Modern information systems can transform observations into increasingly complex representations. A simplified AI pipeline may be expressed as: <math> D_{\mathrm{sensor}} \rightarrow M_{\mathrm{AI}} \rightarrow R_{\mathrm{AI}} \rightarrow H_{\mathrm{human}} </math> where: * <math>D_{\mathrm{sensor}}</math> is sensor data; * <math>M_{\mathrm{AI}}</math> is an AI model or processing system; * <math>R_{\mathrm{AI}}</math> is the resulting representation, classification, prediction, or reconstruction; and * <math>H_{\mathrm{human}}</math> represents human interpretation. AI therefore creates an additional information layer between physical measurement and human understanding. The distinction between raw measurement and processed representation becomes increasingly important when an observation is used to support a claim about an unusual physical event. This produces an important research question: '''How can an observer distinguish information contained in an observation from information introduced by the system interpreting the observation?''' This question does not require time travel. It is already relevant to contemporary AI-assisted image and video analysis. == Augmented Reality and the Information Layer == Augmented reality combines information generated by a digital system with a user's perception of a physical environment. A simplified AR information pathway can be written as: <math> W \rightarrow S \rightarrow D \rightarrow A \rightarrow P \rightarrow H </math> where: * <math>W</math> = physical world; * <math>S</math> = sensors; * <math>D</math> = digital information; * <math>A</math> = augmented-reality overlay; * <math>P</math> = human perception; and * <math>H</math> = human action or interpretation. AR therefore introduces an important distinction between: '''what is physically present''' and: '''what the information system tells the observer about what is physically present'''. The presence of an AR layer would therefore make interpretation more complex, not less. == The AI and AR Feedback System == AI and AR can be considered together as an information-processing system: <math> \text{physical world} \rightarrow \text{sensors} \rightarrow \text{AI processing} \rightarrow \text{digital information} \rightarrow \text{AR display} \rightarrow \text{human perception} \rightarrow \text{human action} </math> Human action can then produce new physical observations. This creates an ordinary feedback structure: <math> \text{information} \rightarrow \text{human action} \rightarrow \text{new observation} \rightarrow \text{new information} </math> Nothing about this ordinary feedback loop requires time travel. It is already a characteristic of many information systems. The significance for the present hypothesis is that the same structure provides a conceptual pathway through which a hypothetical temporal information loop could be represented. == The Time-Loop UAP Hypothesis == The hypothesis can be stated conservatively: '''If a future information-processing system could somehow provide information to an earlier historical period, and if that information influenced observations or actions that later contributed to the future system's own informational state, the resulting process could be modeled as a temporal information loop.''' The hypothesis does not require that the information arrive through a physical spacecraft. It could, in principle, be represented as: * an image; * a measurement; * a prediction; * a coordinate; * a symbolic message; * an AI-generated representation; * an AR overlay; or * another information-bearing signal. The crucial hypothetical property would be its '''temporal provenance'''. The question would not simply be: '''Was a UAP observed?''' Instead, it would be: '''Can the information associated with the observation be demonstrated to have originated from a future state of the same causal system?''' That is a much stronger requirement. No presently established UAP evidence demonstrates this. == Relationship to the Existing Time-Traveler UFO Hypothesis == The present research is related to the broader '''time-traveler UFO hypothesis''' but does not simply propose that UAP are future humans. Instead, it investigates whether a hypothetical '''future-origin information process''' could be represented as a self-consistent temporal information loop involving artificial intelligence (AI), augmented reality (AR), human action, and historical information. The distinction can be expressed as follows: <math> \text{time-traveler UFO hypothesis} \rightarrow \text{UFO/UAP as future human technology} </math> The present research considers a different and more general possibility: <math> \text{future-origin information} \rightarrow \text{past information system} \rightarrow \text{human action} \rightarrow \text{historical information} \rightarrow \text{future information} </math> The present framework therefore does not require that a UAP be a vehicle carrying people from the future. A hypothetical temporal effect could instead involve information, measurement, prediction, artificial intelligence, augmented reality, or another information-processing system. This distinction is important because an '''unidentified observation''' does not by itself establish either extraterrestrial origin or future-human origin. The purpose of the present research is to examine whether the broader concept of temporal information feedback can be formulated mathematically and analyzed using concepts such as causality, closed timelike curves, self-consistency, information provenance, and constrained probability states. The existing time-traveler UFO hypothesis provides a related conceptual background, while the present research develops a more specific information-centered framework. No claim is made that UAP currently provide evidence for future-origin information or temporal information loops. == Temporal Information Feedback == The central proposed feedback structure is: <math> I_{\mathrm{future}} \rightarrow AR_{\mathrm{past}} \rightarrow A_{\mathrm{past}} \rightarrow H_{\mathrm{history}} \rightarrow I_{\mathrm{future}} </math> where: * <math>I_{\mathrm{future}}</math> = hypothetical future-origin information; * <math>AR_{\mathrm{past}}</math> = an augmented-reality or information display in the past; * <math>A_{\mathrm{past}}</math> = human action influenced by that information; and * <math>H_{\mathrm{history}}</math> = the resulting historical record. The resulting information could then become part of the historical data available to the future system. This would constitute a hypothetical '''temporal information feedback loop'''. No such loop has been established experimentally. == From Temporal Information to a UAP Hypothesis == The UAP component of the hypothesis asks whether an apparently anomalous observation could, under a hypothetical temporal-information model, be associated with such a loop. The question is not: '''Are UAPs time travelers?''' Instead, the more limited question is: '''Could an apparently anomalous observation, if future evidence eventually demonstrated that it contained information or technology originating from a future period, be modeled as part of a self-consistent temporal information loop?''' This formulation deliberately postpones the question of origin. An observation could initially remain unexplained while alternative hypotheses are investigated, including: # known aircraft or other human technology; # atmospheric or astronomical phenomena; # sensor artifacts; # viewing geometry; # errors in measurement or interpretation; # image or video processing effects; # incomplete metadata; # conventional technological explanations not initially recognized; and # genuinely unexplained phenomena. A temporal explanation would therefore require substantially stronger evidence than simply observing something unusual. == Physical, Informational, and Interpretive Consistency == The broader research framework distinguishes several types of consistency. === Physical consistency === A hypothetical temporal process would have to satisfy whatever physical laws govern the relevant spacetime. If closed timelike curves were involved, this could include global self-consistency constraints. === Informational consistency === The information appearing at different points in the loop would have to remain mutually compatible. If a future system received a historical record containing information that originated from that future system, the complete information history would need to be examined for consistency. === Interpretive consistency === The interpretation of an observation must remain distinguishable from the underlying measurement. An AI or AR system could alter what an observer perceives without altering the underlying physical event. Therefore: <math> \text{physical event} \neq \text{sensor representation} \neq \text{AI representation} \neq \text{human interpretation} </math> === Ethical consistency === If a future information system could influence historical human decisions, ethical constraints would become relevant. A hypothetical system with access to causally sensitive historical information would raise questions concerning: * information integrity; * unnecessary intervention; * consent; * manipulation; * historical authenticity; * unintended consequences; * accountability; and * preservation of human agency. This connects directly to the broader framework developed in [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. == A Proposed Framework == The following principles are proposed as a conceptual framework. They are not established physical laws. === 1. Information integrity === Measurements, metadata, processing steps, and interpretations should be preserved separately. A future investigation should ideally retain: * original sensor files; * calibration information; * timestamps; * sensor location; * sensor orientation; * exposure information; * instrument characteristics; * processing history; * AI-model information; * transformation records; and * human annotations. === 2. Causal minimalism === If a system had access to causally sensitive information, it should minimize unnecessary intervention. Conceptually: <math> \text{unnecessary intervention} \rightarrow \text{additional causal pathways} \rightarrow \text{greater feedback complexity} </math> In a hypothetical temporal system, minimizing unnecessary intervention could reduce opportunities for unstable or contradictory information loops. === 3. Provenance preservation === Every significant information object should have a traceable origin. For a UAP investigation, this means distinguishing: <math> P_{\mathrm{physical}} \rightarrow P_{\mathrm{sensor}} \rightarrow P_{\mathrm{processing}} \rightarrow P_{\mathrm{AI}} \rightarrow P_{\mathrm{interpretation}} </math> The purpose is to determine which portions of the final representation can be traced to physical measurement and which are products of later processing. === 4. Predictive transparency === A system should distinguish between information that describes an event and information that may influence the event. This distinction becomes important when predictions are shown to people through AI systems or AR displays. The ordinary feedback structure is: <math> \text{prediction} \rightarrow \text{behavior} \rightarrow \text{new data} \rightarrow \text{prediction} </math> A hypothetical temporal system would extend this into: <math> \text{prediction} \rightarrow \text{future action} \rightarrow \text{historical intervention} \rightarrow \text{changed data} \rightarrow \text{prediction} </math> === 5. Self-consistency === If a temporal information loop were ever demonstrated, the complete information history would need to be evaluated rather than examining isolated events. Conceptually: <math> H_{\mathrm{complete}} \in C_{\mathrm{physical}} \cap C_{\mathrm{informational}} \cap C_{\mathrm{interpretive}} \cap C_{\mathrm{ethical}} </math> where the four sets represent different admissibility conditions. This extends the idea of self-consistency into an interdisciplinary framework. == A Possible Temporal AR Loop == A particularly interesting thought experiment involves AR. Imagine that a future information-processing system possesses highly detailed historical information. Suppose, hypothetically, that the system somehow provides a representation of an event to an observer in an earlier period. The observer sees the information through an AR system. The observer then acts on that information. The action becomes part of the historical record. The future system subsequently learns from that historical record. The conceptual loop is: <math> I_F \rightarrow AR_P \rightarrow A_P \rightarrow H \rightarrow AI_F \rightarrow I_F </math> where: * <math>I_F</math> = future information; * <math>AR_P</math> = past AR representation; * <math>A_P</math> = past action; * <math>H</math> = historical record; and * <math>AI_F</math> = future AI system. This is a thought experiment. It does not establish that such a mechanism exists. == Bootstrap Information and the Origin Problem == A classic information-loop problem arises when information exists in a causal loop without an identifiable external origin. For example: <math> I_1 \rightarrow I_2 \rightarrow I_3 \rightarrow I_1 </math> If the information contained in <math>I_1</math> is completely determined by later information in the same loop, it may become difficult to identify an independent source for the information. This does not automatically constitute a contradiction. However, it creates a major epistemological problem. A scientific investigation would need to distinguish: * information genuinely originating from an external source; * information generated internally by the system; * information reconstructed from incomplete observations; * information introduced by human interpretation; * information produced by AI systems; and * information that only appears to have a future origin because its provenance is incomplete. The bootstrap problem therefore makes provenance especially important. == UAP Observation Versus UAP Interpretation == The framework proposes maintaining separate levels of description. '''Level 1 — Observation''' What was directly measured or recorded? '''Level 2 — Description''' What does the recording objectively show? '''Level 3 — Processing''' What transformations, enhancement, classification, or reconstruction were applied? '''Level 4 — Interpretation''' What explanations are consistent with the available evidence? '''Level 5 — Hypothesis''' What broader models could potentially explain the observation? A temporal-loop explanation belongs at Level 5. It should not be inserted into Level 1 or Level 2 merely because an observation appears unusual. == Relationship to Augmented Reality == AR introduces an additional methodological problem because the displayed scene can contain both physical and digital information. Conceptually: <math> R_{\mathrm{display}} = R_{\mathrm{physical}} + I_{\mathrm{digital}} </math> where: * <math>R_{\mathrm{display}}</math> represents the displayed perceptual environment; * <math>R_{\mathrm{physical}}</math> represents information derived from the physical environment; and * <math>I_{\mathrm{digital}}</math> represents digitally generated or retrieved information. This equation is a conceptual representation rather than a physical law. If the digital information were generated by an AI system, the observer could potentially perceive an object, annotation, prediction, reconstruction, or other representation that is not identical to a directly observed physical object. Therefore, an AR-mediated observation would require careful preservation of the underlying sensor data and metadata before conclusions were drawn about the physical phenomenon. == Information Provenance and Historical Integrity == The hypothesis requires unusually strong attention to provenance. For an information object <math>I</math>, provenance can be represented conceptually as: <math> \Pi(I) = \{t,s,m,p,a,r\} </math> where: * <math>t</math> = timestamp; * <math>s</math> = source; * <math>m</math> = metadata; * <math>p</math> = processing history; * <math>a</math> = analytical actions; and * <math>r</math> = resulting representation. A temporal-origin claim would require showing that the provenance chain cannot be explained by ordinary forward information flow. This is a much stronger requirement than demonstrating that a prediction happened to be correct. == What Would Count as Evidence? == The Time-Loop UAP Hypothesis should require substantially stronger evidence than an unusual image or unexplained observation. Potentially relevant evidence would include: === 1. Independent physical measurement === The phenomenon should be recorded by calibrated instruments capable of producing quantitative measurements. === 2. Multiple synchronized observations === Independent sensors should observe the same event with synchronized timing. === 3. Complete metadata === The record should include sufficient information to reconstruct the observational circumstances. === 4. Preservation of original data === Original sensor data should be preserved rather than relying exclusively on screenshots, compressed video, or later reconstructions. === 5. Independent analysis === The observation should be analyzable by researchers who are not dependent upon the original interpretation. === 6. Demonstrable information provenance === If future-origin information is proposed, the information should have a demonstrable provenance that cannot be explained by ordinary information transfer. === 7. Temporal priority === The allegedly future-origin information would need to be demonstrably present before the event or information from which it would ordinarily have been derived. This is crucial. A prediction generated after an event cannot ordinarily establish that the information originated from the future. === 8. Reproducibility === The effect should be observable or testable independently. === 9. Falsifiable predictions === The hypothesis should produce predictions that distinguish it from competing explanations. === 10. Exclusion of conventional explanations === Sensor artifacts, atmospheric effects, astronomical objects, viewing geometry, image processing, human technology, and other ordinary explanations should be investigated before a temporal explanation is considered. == Alternative Explanations and Falsifiability == The hypothesis should remain subordinate to ordinary explanations until evidence requires otherwise. Potential explanations include: * conventional aircraft or technology; * atmospheric or astronomical phenomena; * sensor artifacts; * image-processing effects; * viewing-geometry effects; * AI interpretation; * AR-generated information; * incomplete metadata; * conventional informational feedback; and * genuinely unexplained phenomena. A temporal explanation would require evidence that distinguishes it from these alternatives. An unresolved observation therefore remains an unresolved observation until additional evidence establishes a specific explanation. == Mathematical Formulation == The equations in this section are proposed mathematical representations of the conceptual framework. They are not equations derived from quantum mechanics or established models of time travel. === State-transition representation === Let <math>S_t</math> represent the information state of a system at time <math>t</math>. An ordinary forward process can be represented as: <math> S_{t+1}=F_t(S_t,A_t) </math> where <math>A_t</math> represents actions or interventions at time <math>t</math>. A hypothetical temporally coupled system could be represented as: <math> S_t=G_t(S_{t-1},A_{t-1},S_{t+1},B_t) </math> where <math>B_t</math> represents additional boundary or environmental information. The future-state dependence is the speculative element. === Trajectory and fixed-point representation === A complete trajectory may be written: <math> S=(S_0,S_1,\ldots,S_T) </math> A self-consistent trajectory can be represented by: <math> S^{*}=\Phi(S^{*}) </math> This means that the complete history reproduces itself under the transformation and constraints represented by <math>\Phi</math>. This is a mathematical analogy rather than an established physical law. === Physical, informational, interpretive, and ethical admissibility === The proposed framework can be represented as: <math> S^{*} \in C_{\mathrm{physical}} \cap C_{\mathrm{informational}} \cap C_{\mathrm{interpretive}} \cap C_{\mathrm{ethical}} </math> A history would therefore be considered admissible in the model only if it satisfies all four classes of constraint. == Probability-Based State Transformations == A separate research question concerns whether constrained probability transformations can provide useful mathematical analogies for temporally constrained systems. Let: <math> p_t(x)\geq0 </math> and: <math> \sum_x p_t(x)=1 </math> where <math>p_t(x)</math> represents the probability of information state <math>x</math> at time <math>t</math>. A probabilistic transformation can be written as: <math> \widetilde{p}_{t+1}(x) = T_t(p_t,x) </math> If the transformed distribution is not normalized, define: <math> Z_{t+1} = \sum_x \widetilde{p}_{t+1}(x) </math> and: <math> p_{t+1}(x) = \frac{\widetilde{p}_{t+1}(x)} {Z_{t+1}} </math> This provides an abstract analogy: '''transformation → constraint → admissible state''' However, probability normalization alone is not equivalent to temporal consistency. Normalization ensures that a probability distribution sums to one. It does not ensure that a corresponding history is physically possible, informationally coherent, or ethically permissible. A trajectory-level admissibility condition can be represented as: <math> \mathcal{A}_{\mathrm{total}} = \mathcal{A}_{\mathrm{phys}} \cap \mathcal{A}_{\mathrm{info}} \cap \mathcal{A}_{\mathrm{interp}} \cap \mathcal{A}_{\mathrm{eth}} </math> where: * <math>\mathcal{A}_{\mathrm{phys}}</math> represents physically admissible trajectories; * <math>\mathcal{A}_{\mathrm{info}}</math> represents informationally consistent trajectories; * <math>\mathcal{A}_{\mathrm{interp}}</math> represents interpretation-consistent trajectories; and * <math>\mathcal{A}_{\mathrm{eth}}</math> represents ethically admissible trajectories. A constrained probability distribution could then be written conceptually as: <math> P_{\mathrm{adm}}(x) = P(x\mid x\in\mathcal{A}_{\mathrm{total}}) </math> This remains an abstract modeling analogy. It does not imply that ethical rules are physical laws or that quantum mechanics requires the proposed ethical constraints. == Relationship to Probability Dilation Theory == The probability-based formulation described above has a potential connection with '''Probability Dilation Theory (PDT)''', an independent mathematical research framework developed as part of the broader research project. PDT studies transformations of probability distributions under dilation and normalization. A basic PDT transformation can be represented as: <math> T_M(P)=\frac{MP}{\lVert MP\rVert_1} </math> where <math>P</math> is a probability vector and <math>M</math> is a transformation matrix. The normalization condition is: <math> \sum_i P_i=1 </math> This provides a mathematical framework for studying how a probability state changes while remaining within the probability simplex. The connection to the present hypothesis is '''conceptual and exploratory'''. The Time-Loop UAP Hypothesis considers whether a hypothetical temporal information system could involve constrained information states and self-consistent trajectories. PDT provides one possible mathematical framework for studying transformations of probability states within such a system. This does '''not''' imply that PDT predicts time travel, closed timelike curves, UAP, or future-origin information. Instead, a possible research question is: '''Can PDT's probability-state transformations, normalization procedures, and fixed-point structures provide a useful mathematical representation of constrained information states in a hypothetical temporal information loop?''' A possible conceptual relationship is: <math> \text{PDT probability state} \rightarrow \text{information transformation} \rightarrow \text{constraint} \rightarrow \text{admissible state} </math> This can be compared structurally with the temporal-consistency formulation: <math> S^{*}=\Phi(S^{*}) </math> where <math>S^{*}</math> represents a self-consistent state or trajectory. The two formulations should not be assumed to be mathematically equivalent. The similarity is currently a research question rather than an established result. Future work could investigate whether PDT fixed points, dilation flows, Fisher geometry, or other mathematical structures developed within PDT have meaningful relationships with self-consistent information trajectories. This would provide a possible bridge between the mathematical development of PDT and the broader study of information, causality, and temporal consistency. The distinction between '''mathematical correspondence''' and '''physical interpretation''' should be maintained throughout this research. == PDT Research Questions == The following questions may be investigated in future work: # Can PDT transformations be formulated on a state space representing temporally constrained information? # Do PDT fixed points have a useful interpretation as self-consistent probability states? # Can PDT dilation flows be used to model convergence toward an admissible information state? # What quantities are preserved or changed under PDT normalization? # Is there a meaningful relationship between PDT geometry and the geometry of a temporally constrained state space? # Can Fisher information or Fisher–Rao geometry provide additional mathematical constraints on PDT trajectories? # Can PDT produce experimentally distinguishable predictions in any physical application? # What mathematical conditions would be required before a PDT model could be interpreted as more than an analogy to temporal self-consistency? At the present stage, these questions remain open. PDT should therefore be regarded as a '''potential mathematical research connection''' rather than as evidence for the Time-Loop UAP Hypothesis. == Relationship to Closed Timelike Curves == The hypothesis draws conceptual inspiration from research concerning closed timelike curves. In a spacetime containing a closed timelike curve, a worldline can in principle return to an earlier spacetime event. The important theoretical issue is whether a complete solution can remain self-consistent. Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever examined the Cauchy problem in spacetimes containing closed timelike curves and discussed the need for a global self-consistency condition if such curves are permitted. Echeverria, Klinkhammer, and Thorne subsequently examined a classical billiard-ball model involving wormhole-generated closed timelike curves and found examples involving multiple self-consistent solutions. The Time-Loop UAP Hypothesis does not assert that UAP observations involve closed timelike curves. Instead, closed-timelike-curve research supplies a theoretical vocabulary for asking how a causal loop might remain globally consistent. == Relationship to the Novikov Self-Consistency Principle == The [[Novikov Self-Consistency Principle]] provides an especially relevant conceptual connection. The principle can be summarized as: <math> \text{local possibility} \rightarrow \text{global consistency requirement} \rightarrow \text{admissible history} </math> Applied only as a speculative analogy, the principle suggests that if a temporal information loop existed, not every apparently possible information sequence would necessarily correspond to an admissible history. The hypothesis therefore does not assume: '''The past can be changed arbitrarily.''' Instead it asks: '''If future-origin information existed, what global constraints would be required for the complete information history to remain self-consistent?''' This is a substantially narrower proposition. == Relationship to Temporal Consistency and Quantum Ethics == The strongest conceptual relationship is with [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]]. The main article identifies a progression: <math> \text{physical consistency} \rightarrow \text{informational consistency} \rightarrow \text{ethical consistency} </math> The present UAP hypothesis adds another layer: <math> \text{physical observation} \rightarrow \text{information} \rightarrow \text{interpretation} \rightarrow \text{action} \rightarrow \text{historical information} </math> The two frameworks can therefore be combined conceptually. A hypothetical future AI system might not need to physically transport an object through time. It would only need, in the speculative model, to affect information that later becomes part of its own historical input. This produces the possibility of: <math> \text{information} \rightarrow \text{action} \rightarrow \text{history} \rightarrow \text{future information} </math> The main article already considers a related structure involving augmented reality: '''AR information → human behavior → historical record → future training data''' The present article extends that conceptual structure into the UAP domain. The important distinction remains: '''ordinary informational feedback is already real; a temporal feedback loop remains hypothetical.''' == Ethical Implications of a Hypothetical Temporal Information System == If a future information-processing system could affect historical information, the ethical problem would extend beyond physics. A hypothetical system might face questions such as: * Should it intervene at all? * How much intervention would be justified? * Should it disclose the origin of information? * Could historical people give meaningful consent? * What happens if the information changes subsequent decisions? * Who is responsible for unintended consequences? * How should historical records be protected from deliberate manipulation? * Should an AI system be permitted to optimize outcomes by altering information conditions? These questions are consistent with the broader concept of '''ethical consistency''' developed in the main research article. The ethical framework does not depend on proving time travel. Similar questions already arise when AI systems influence human behavior and subsequently learn from the resulting behavior. == Science Fiction and Conceptual Analogies == Science fiction can provide useful conceptual analogies, but it should not be treated as scientific evidence. The broader research project uses [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]] to examine fictional representations of: * advanced AI; * information asymmetry; * autonomous decision-making; * objectives and constraints; * human-machine interaction; and * consequences of information systems. The present article does not claim that fictional depictions of UFOs or time travel predict physical reality. Fiction can instead serve as a conceptual laboratory in which questions concerning causality, information, AI, and temporal consistency can be examined. == Research Questions == The following questions remain open within the speculative framework: # Can UAP investigation methods be designed to preserve complete sensor and processing provenance? # How can AI-generated interpretations be distinguished from direct sensor measurements? # How should AR systems distinguish physical observations from digital overlays? # Can information-feedback loops be modeled using established causal and information-theoretic mathematics? # Under what assumptions can a temporal information loop be represented as a fixed-point problem? # What conditions would be required for a future-origin information claim to be empirically testable? # Could a future information system create a logically self-consistent information loop without producing an observable contradiction? # Would such a loop necessarily involve a physical closed timelike curve, or could a different theoretical mechanism produce an equivalent information structure? # How should bootstrap information be treated within a causal model? # What would distinguish a genuine temporal information effect from an ordinary prediction, coincidence, data leak, or reconstruction? # Could probability-based state transformations provide a useful mathematical language for studying constrained temporal information systems? # What ethical constraints would be appropriate for an information-processing system capable of influencing historical conditions? == Limitations and Open Questions == This article has substantial limitations. First, no established experiment demonstrates controllable communication from the future into the past. Second, no established evidence demonstrates that UAP are caused by time travel. Third, unexplained observations do not by themselves establish a new physical mechanism. Fourth, AI and AR can introduce additional layers of interpretation between a physical event and human perception. Fifth, the mathematical formulations in this article are proposed models and analogies rather than experimentally validated physical equations. Sixth, a self-consistent mathematical model does not demonstrate that the modeled process occurs in nature. Seventh, a correct prediction does not by itself establish future-origin information. Finally, the hypothesis may ultimately prove unfalsifiable or physically impossible if no mechanism exists by which future-origin information can influence past observations. These limitations are not weaknesses to be hidden; they are part of the scientific definition of the hypothesis. == Conclusion == The Time-Loop UAP Hypothesis proposes a conservative question rather than a conclusion. UAP are treated as observations rather than explanations. The research begins with a standard information pathway: <math> \text{physical event} \rightarrow \text{sensor} \rightarrow \text{data} \rightarrow \text{processing} \rightarrow \text{interpretation} </math> AI and AR introduce additional information-processing layers. An ordinary feedback system can then arise: <math> \text{prediction} \rightarrow \text{human action} \rightarrow \text{new data} \rightarrow \text{new prediction} </math> The speculative extension is: <math> \text{future-origin information} \rightarrow \text{past information system} \rightarrow \text{human action} \rightarrow \text{historical record} \rightarrow \text{future information} </math> If such a process were ever demonstrated, the concepts of physical consistency, informational consistency, interpretive integrity, provenance, and ethical consistency could provide a framework for analyzing it. The hypothesis therefore does not begin with the proposition that UAP are time travelers. It begins with a more general scientific question: '''Could future-origin information, if independently demonstrated, form part of a globally self-consistent information loop?''' At present, this remains a speculative research question. == References == === Physics and temporal consistency === * Friedman, J. L., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., and Yurtsever, U. (1990). "Cauchy problem in spacetimes with closed timelike curves." ''Physical Review D'', 42, 1915–1930. [https://doi.org/10.1103/PhysRevD.42.1915 DOI] * Echeverria, F., Klinkhammer, G., and Thorne, K. S. (1991). "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory." ''Physical Review D'', 44, 1077–1099. [https://doi.org/10.1103/PhysRevD.44.1077 DOI] * Hawking, S. W. (1992). "Chronology protection conjecture." ''Physical Review D'', 46, 603–611. [https://doi.org/10.1103/PhysRevD.46.603 DOI] * Deutsch, D. (1991). "Quantum mechanics near closed timelike lines." ''Physical Review D'', 44, 3197–3217. [https://doi.org/10.1103/PhysRevD.44.3197 DOI] === UAP methodology === * NASA. (2023). ''NASA Unidentified Anomalous Phenomena Independent Study Team Final Report''. [https://science.nasa.gov/uap/ NASA UAP Independent Study] === Artificial intelligence and information systems === * National Institute of Standards and Technology. (2023). ''Artificial Intelligence Risk Management Framework (AI RMF 1.0)''. [https://www.nist.gov/itl/ai-risk-management-framework NIST AI Risk Management Framework] * National Institute of Standards and Technology. ''AI Measurement and Evaluation''. [https://www.nist.gov/ai-measurement-and-evaluation NIST AI Measurement and Evaluation] === Augmented reality === * Azuma, R. T. (1997). "A Survey of Augmented Reality." ''Presence: Teleoperators and Virtual Environments'', 6(4), 355–385. [https://doi.org/10.1162/pres.1997.6.4.355 DOI] === Fiction and artificial intelligence === * Kubrick, S., director. (1968). ''2001: A Space Odyssey''. Metro-Goldwyn-Mayer. == Further Reading == * [[Closed Timelike Curves]] * [[Novikov Self-Consistency Principle]] * [[Causality]] * [[Temporal Consistency and Quantum Ethics: 2001 as a Fictional Analogue]] * [https://science.nasa.gov/uap/ NASA UAP Independent Study] * [https://www.nist.gov/itl/ai-risk-management-framework NIST AI Risk Management Framework] * [https://doi.org/10.1162/pres.1997.6.4.355 Azuma, "A Survey of Augmented Reality"] == Research Status == '''Research status: Speculative interdisciplinary framework.''' Version 1.1 develops a conceptual hypothesis connecting UAP observation, information provenance, artificial intelligence, augmented reality, temporal information feedback, causal consistency, and ethical constraints. The hypothesis does not claim that UAP are extraterrestrial spacecraft, time travelers, or evidence of backward time travel. No established scientific experiment currently demonstrates controllable information transfer from the future to the past. The mathematical formulations presented in this article are proposed models and analogies for further investigation. Future versions may refine the mathematical framework, develop computational simulations, examine information-theoretic constraints, and investigate whether the proposed temporal-information structures can produce experimentally distinguishable predictions. == Author and Attribution == This research article is part of an ongoing independent research project developed by Howard Richardson. The ideas presented here are exploratory and are intended to be developed, criticized, tested, refined, or rejected through further research. The publication of a speculative hypothesis on Wikiversity does not constitute scientific confirmation. The article is released for educational and research discussion under the applicable Wikiversity and Wikimedia licensing framework. tu1k3uen3sd6v6uwa2lf0ga8uch1vcf 3 332313 2835500 2026-10-02T12:21:33Z MathXplore 2888076 advert1 ([[m:User:ZbVl/VD|Vandoom]]) 2835500 wikitext text/x-wiki == 2026-10-02 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Isiaka barakat|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using <span style="white-space:nowrap">Wikiversity</span> as a "soapbox"]] are not permitted. Take a look at the welcome pages to learn more about <span style="white-space:nowrap">Wikiversity</span>. Thanks. </div><!-- Glow-advert1 @ 1790943673322.5s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:21, 2 October 2026 (UTC) ide3rw6a2rl05iira9dtfs7d59dvenz 3 332314 2835501 2026-10-02T12:23:38Z MathXplore 2888076 advert1 ([[m:User:ZbVl/VD|Vandoom]]) 2835501 wikitext text/x-wiki == 2026-10-02 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/Cunningdentalgroupca|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using <span style="white-space:nowrap">Wikiversity</span> as a "soapbox"]] are not permitted. Take a look at the welcome pages to learn more about <span style="white-space:nowrap">Wikiversity</span>. Thanks. </div><!-- Glow-advert1 @ 1790943816774.5s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:23, 2 October 2026 (UTC) ojrh3rle086420yximng9cofkazp8on 0 332315 2835520 2026-10-02T15:19:03Z ~2026-53120-09 3112000 Created page with "Map Skills Introduction Map skills help learners understand where places are located and how different geographic features relate to one another. These skills are useful for studying countries, regions, cities, oceans, physical features, and patterns shown on maps. This learning resource introduces basic map-reading and country-recognition skills through simple activities. It is designed for beginners and can be used for independent study or classroom practice. Learn..." 2835520 wikitext text/x-wiki Map Skills Introduction Map skills help learners understand where places are located and how different geographic features relate to one another. These skills are useful for studying countries, regions, cities, oceans, physical features, and patterns shown on maps. This learning resource introduces basic map-reading and country-recognition skills through simple activities. It is designed for beginners and can be used for independent study or classroom practice. Learning objectives By the end of this resource, learners should be able to: • identify broad regions and countries on a world map • use neighbouring countries, coastlines, oceans, and relative location as geographic clues • describe basic relationships between places shown on a map • practise country recognition using active recall • use interactive activities to reinforce map knowledge 1. Understanding World Maps A world map provides a visual representation of the Earth's surface. Instead of learning country names as an isolated list, learners can connect each country with its location, neighbouring countries, surrounding oceans, and broader geographic region. Start by looking at a world map and identifying the major regions of the world: North America, South America, Europe, Africa, Asia, and Oceania. Antarctica can also be identified as a distinct polar region. Try to notice how countries are positioned in relation to one another. These relationships can provide useful clues when identifying an unfamiliar country. 2. Recognising Countries and Regions Country recognition becomes easier when names are connected with visual and geographic information. Choose one region at a time and study a small group of countries. Pay attention to their approximate location, shape, neighbours, coastlines, and nearby bodies of water. After studying the map, look away and try to recall where the countries are located. Check the map afterwards and review the locations that were difficult to remember. 3. Using Geographic Clues Maps provide several clues that can help identify locations. Useful clues include: • neighbouring countries • coastlines • oceans and seas • relative position within a region • distinctive country shapes • nearby geographic features For example, rather than trying to remember a country completely on its own, consider which countries surround it and whether it has a coastline. Connecting several clues can make location easier to recall. 4. Active Recall for Map Learning Active recall means trying to remember information before checking the answer or your notes. A simple practice routine is: 1. Study a small area of a map. 2. Look away from the map. 3. Try to name or locate the countries you studied. 4. Check your answers. 5. Review the locations you found difficult. 6. Repeat the activity later. The purpose is not to memorise an entire world map in one session. Short, repeated attempts can help strengthen familiarity with geographic locations. 5. Country Recognition Practice Choose a region that you have recently studied. Without looking at your notes, try to name several countries from that region and recall where they are located. Then check your answers against a map. For additional practice, learners can use an interactive countries-of-the-world quiz that presents map-based country recognition questions. After completing the activity, review the countries that were difficult to identify and return to the map to reinforce their locations. 6. Review Activity Choose one region of the world and complete the following: • Identify five countries in the region. • Identify at least two neighbouring-country relationships. • Name a nearby ocean or sea where applicable. • Try to locate the countries without looking at your notes. • Check your answers and review any mistakes. Repeat the activity with another region when you are ready. Conclusion Map skills develop through observation, comparison, recall, and repeated practice. By connecting country names with locations and geographic relationships, learners can gradually become more comfortable using world maps. The goal is to understand geographic relationships rather than rely only on memorising lists. 0d8skyl8pwqdazrudgb35uznn5fmofr 2835522 2835520 2026-10-02T15:31:37Z ~2026-53120-09 3112000 2835522 wikitext text/x-wiki '''Map Skills''' '''Introduction''' Map skills help learners understand where places are located and how different geographic features relate to one another. These skills are useful for studying countries, regions, cities, oceans, physical features, and patterns shown on maps. This learning resource introduces basic map-reading and country-recognition skills through simple activities. It is designed for beginners and can be used for independent study or classroom practice. '''Learning objectives''' By the end of this resource, learners should be able to: • identify broad regions and countries on a world map • use neighboring countries, coastlines, oceans, and relative location as geographic clues • describe basic relationships between places shown on a map • practice country recognition using active recall • use interactive activities to reinforce map knowledge '''1. Understanding World Maps''' A world map provides a visual representation of the Earth's surface. Instead of learning country names as an isolated list, learners can connect each country with its location, neighbouring countries, surrounding oceans, and broader geographic region. Start by looking at a world map and identifying the major regions of the world: North America, South America, Europe, Africa, Asia, and Oceania. Antarctica can also be identified as a distinct polar region. Try to notice how countries are positioned in relation to one another. These relationships can provide useful clues when identifying an unfamiliar country. '''2. Recognizing Countries and Regions''' Country recognition becomes easier when names are connected with visual and geographic information. Choose one region at a time and study a small group of countries. Pay attention to their approximate location, shape, neighbors, coastlines, and nearby bodies of water. After studying the map, look away and try to recall where the countries are located. Check the map afterwards and review the locations that were difficult to remember. '''3. Using Geographic Clues''' Maps provide several clues that can help identify locations. Useful clues include: • neighboring countries • coastlines • oceans and seas • relative position within a region • distinctive country shapes • nearby geographic features For example, rather than trying to remember a country completely on its own, consider which countries surround it and whether it has a coastline. Connecting several clues can make location easier to recall. '''4. Active Recall for Map Learning''' Active recall means trying to remember information before checking the answer or your notes. A simple practice routine is: 1. Study a small area of a map. 2. Look away from the map. 3. Try to name or locate the countries you studied. 4. Check your answers. 5. Review the locations you found difficult. 6. Repeat the activity later. The purpose is not to memorize an entire world map in one session. Short, repeated attempts can help strengthen familiarity with geographic locations. '''5. Country Recognition Practice''' Choose a region that you have recently studied. Without looking at your notes, try to name several countries from that region and recall where they are located. Then check your answers against a map. For additional practice, learners can use an [https://www.geomapgame.com/countries-of-the-world-quiz interactive countries-of-the-world quiz] that presents map-based country recognition questions. After completing the activity, review the countries that were difficult to identify and return to the map to reinforce their locations. '''6. Review Activity''' Choose one region of the world and complete the following: • Identify five countries in the region. • Identify at least two neighboring-country relationships. • Name a nearby ocean or sea where applicable. • Try to locate the countries without looking at your notes. • Check your answers and review any mistakes. Repeat the activity with another region when you are ready. '''Conclusion''' Map skills develop through observation, comparison, recall, and repeated practice. By connecting country names with locations and geographic relationships, learners can gradually become more comfortable using world maps. The goal is to understand geographic relationships rather than rely only on memorizing lists. or71zea0pxaa2he88kzkxg8fidheug0 0 332316 2835523 2026-10-02T15:53:12Z Dick Bos 24466 first lines 2835523 wikitext text/x-wiki {{10% done}} Welcome to the pages about '''William Petty'''. William Petty (1623–1687) was an English scientist, best known for his contributions to the very early fields of political economy and statistics. As a young man, he had a number of remarkable adventures and, in a highly unconventional manner and within a short space of time, managed to acquire knowledge in a wide range of fields. Around 1651, he worked as a lecturer in anatomy at Brasenose College, Oxford, and was also appointed Gresham Professor of Music in London. Just one year later, he set off for Ireland, following in the footsteps of Cromwell’s army. In 1654, he secured a contract to map Ireland: the Down Survey. He acquired thousands of hectares of land in south-west Ireland. Meanwhile, he also carried out all manner of research and was active as an inventor. His economic work was decisively influenced by [[:w:en:Thomas Hobbes|Thomas Hobbes]]. Petty served as his personal secretary for some time. [[:w:en:Francis Bacon|Francis Bacon]]’s influence on the development of Petty’s thinking was also significant. Petty's first important economic publication{{sfn|Schumpeter|1981|p=210}} was ''A Treatise of Taxes and Contributions'' (1662). This was soon followed by a text, written 1665, only published 1691: ''Verbum Sapienti''. His ''Political Anatomy of Ireland'' followed 1672. In 1676 he wrote ''Political Arithmetick'' (first published 1690). His ''Quantulumcunque concerning Money'' was written 1682. Important were his ''Essays on Political Arithmetick'' (written between 1671 and 1687).<br />All these texts were republished by Charles H. Hull in 1899 in ''The Economic Writings of Sir William Petty''. And then there is ''Natural and Political Observations … upon the Bills of Mortality,'' published in 1662 by John Graunt. But the influence of Petty on this text has been considerable, to say the least. == Footnotes == {{References}} == Sources == * {{cite book | last = Schumpeter | first = Joseph A. | date = 1981 | orig-date = 1951 | editor-last = Schumpter | editor-first = Elizabeth Boody | title = History of Economic Analysis | location = London etc. | publisher = George Allen & Unwin | ISBN = 0 04 330086 3 | ref = harv }} [[category:History of Economic Thought]] rgzrxr684k8vsr03pdq9cpze76sr1oo 0 332317 2835527 2026-10-02T17:05:58Z ~2026-53063-77 3112004 Subpage for Causality 2835527 wikitext text/x-wiki == Causality == Causality is central to the study of temporal consistency because physical events are ordinarily understood to occur within a causal structure in which causes precede their effects. In relativity, the causal relationship between events is constrained by the structure of spacetime and the light cone. General relativity also permits theoretical solutions in which unusual causal structures, including closed timelike curves, can arise. The present research does not assume that retrocausality or closed timelike curves occur in nature. Instead, these concepts provide a theoretical framework for examining whether a hypothetical temporal information system could remain causally and informationally self-consistent. For a more detailed discussion of causality, including causal structure, relativity, causal loops, closed timelike curves, the Novikov Self-Consistency Principle, chronology protection, and information, see the companion Wikiversity page: 6cw4r1nh45c4zc4t64s54hnzxuu655t 2835657 2835527 2026-10-03T08:59:02Z Michael Ten 654933 removed extra duplicate heading 2835657 wikitext text/x-wiki Causality is central to the study of temporal consistency because physical events are ordinarily understood to occur within a causal structure in which causes precede their effects. In relativity, the causal relationship between events is constrained by the structure of spacetime and the light cone. General relativity also permits theoretical solutions in which unusual causal structures, including closed timelike curves, can arise. The present research does not assume that retrocausality or closed timelike curves occur in nature. Instead, these concepts provide a theoretical framework for examining whether a hypothetical temporal information system could remain causally and informationally self-consistent. For a more detailed discussion of causality, including causal structure, relativity, causal loops, closed timelike curves, the Novikov Self-Consistency Principle, chronology protection, and information, see the companion Wikiversity page: 9ic5ce32cj1jegia80do1a3rgkfuz3h 0 332318 2835528 2026-10-02T17:09:16Z Howie2024 2995240 Novikov self-consistency subpage 2835528 wikitext text/x-wiki == Novikov Self-Consistency Principle == The '''Novikov Self-Consistency Principle''' provides a theoretical framework for considering how events might remain logically consistent in a spacetime containing closed timelike curves. The principle proposes, in simplified form, that physical processes occurring within a closed timelike curve must be globally self-consistent. Events cannot produce a contradiction within the complete causal history. This concept is relevant to the present research because a hypothetical temporal information system would need to satisfy similar questions of consistency if information were able to influence events in its own causal history. The present research does not assume that closed timelike curves or backward time travel exist physically. The Novikov principle is used here as a theoretical framework for examining the broader concept of '''temporal self-consistency'''. For a more detailed discussion of the Novikov Self-Consistency Principle, including its relationship to closed timelike curves, causal loops, information, multiple self-consistent histories, and artificial intelligence, see the companion Wikiversity page: edc30vy5cqqin4f0e9cngtt4acvt2aa 2835658 2835528 2026-10-03T08:59:10Z Michael Ten 654933 2835658 wikitext text/x-wiki The '''Novikov Self-Consistency Principle''' provides a theoretical framework for considering how events might remain logically consistent in a spacetime containing closed timelike curves. The principle proposes, in simplified form, that physical processes occurring within a closed timelike curve must be globally self-consistent. Events cannot produce a contradiction within the complete causal history. This concept is relevant to the present research because a hypothetical temporal information system would need to satisfy similar questions of consistency if information were able to influence events in its own causal history. The present research does not assume that closed timelike curves or backward time travel exist physically. The Novikov principle is used here as a theoretical framework for examining the broader concept of '''temporal self-consistency'''. For a more detailed discussion of the Novikov Self-Consistency Principle, including its relationship to closed timelike curves, causal loops, information, multiple self-consistent histories, and artificial intelligence, see the companion Wikiversity page: nwj86kedgtg4de4dkt83w2g2isdg6yy 0 332319 2835530 2026-10-02T17:15:28Z Howie2024 2995240 Chronology Protection Conjecture subpage. 2835530 wikitext text/x-wiki == Chronology Protection Conjecture == The '''Chronology Protection Conjecture''' is a proposal associated with Stephen Hawking that the laws of physics may prevent the formation of closed timelike curves and therefore protect the causal structure of spacetime from the appearance of macroscopic time machines. The conjecture is relevant to the present research because it provides an important alternative to models in which temporal loops are physically realizable. If chronology protection were a fundamental feature of nature, hypothetical backward-time information processes would face a physical constraint that would need to be accounted for. The present research does not assume that chronology protection has been established as a complete physical law. Instead, it is considered as one of several theoretical perspectives on the possibility of closed timelike curves and temporal feedback. This distinction is important for the broader research framework: '''temporal self-consistency''' and '''chronology protection''' address related but different questions. A self-consistency model asks how a temporal loop could remain logically consistent if such a loop existed, whereas chronology protection considers whether physical laws might prevent such loops from forming in the first place. For a more detailed discussion of chronology protection, its relationship to closed timelike curves, quantum effects, and the broader problem of time travel in physics, see the companion Wikiversity page: '''[[Chronology Protection Conjecture]]''' 1vad9aztarxf6mecjncw72f6gcu0f44 2835531 2835530 2026-10-02T17:16:22Z Howie2024 2995240 Minor link repair. 2835531 wikitext text/x-wiki == Chronology Protection Conjecture == The '''Chronology Protection Conjecture''' is a proposal associated with Stephen Hawking that the laws of physics may prevent the formation of closed timelike curves and therefore protect the causal structure of spacetime from the appearance of macroscopic time machines. The conjecture is relevant to the present research because it provides an important alternative to models in which temporal loops are physically realizable. If chronology protection were a fundamental feature of nature, hypothetical backward-time information processes would face a physical constraint that would need to be accounted for. The present research does not assume that chronology protection has been established as a complete physical law. Instead, it is considered as one of several theoretical perspectives on the possibility of closed timelike curves and temporal feedback. This distinction is important for the broader research framework: '''temporal self-consistency''' and '''chronology protection''' address related but different questions. A self-consistency model asks how a temporal loop could remain logically consistent if such a loop existed, whereas chronology protection considers whether physical laws might prevent such loops from forming in the first place. For a more detailed discussion of chronology protection, its relationship to closed timelike curves, quantum effects, and the broader problem of time travel in physics, see the companion Wikiversity page: h9ghyzy678xb4u1y71uihu3scm5pl2b 2835582 2835531 2026-10-02T22:43:59Z MathXplore 2888076 added [[Category:Conjectures]] using [[Help:Gadget-HotCat|HotCat]] 2835582 wikitext text/x-wiki == Chronology Protection Conjecture == The '''Chronology Protection Conjecture''' is a proposal associated with Stephen Hawking that the laws of physics may prevent the formation of closed timelike curves and therefore protect the causal structure of spacetime from the appearance of macroscopic time machines. The conjecture is relevant to the present research because it provides an important alternative to models in which temporal loops are physically realizable. If chronology protection were a fundamental feature of nature, hypothetical backward-time information processes would face a physical constraint that would need to be accounted for. The present research does not assume that chronology protection has been established as a complete physical law. Instead, it is considered as one of several theoretical perspectives on the possibility of closed timelike curves and temporal feedback. This distinction is important for the broader research framework: '''temporal self-consistency''' and '''chronology protection''' address related but different questions. A self-consistency model asks how a temporal loop could remain logically consistent if such a loop existed, whereas chronology protection considers whether physical laws might prevent such loops from forming in the first place. For a more detailed discussion of chronology protection, its relationship to closed timelike curves, quantum effects, and the broader problem of time travel in physics, see the companion Wikiversity page: [[Category:Conjectures]] juy78jamgcv4radgngpgkswxm3zkh6l 0 332320 2835532 2026-10-02T17:18:48Z Howie2024 2995240 Two-State Vector Formalism 2835532 wikitext text/x-wiki == Two-State Vector Formalism == The '''Two-State Vector Formalism''' (TSVF) is a time-symmetric formulation of quantum mechanics associated with Aharonov and collaborators. Rather than describing a quantum system solely in terms of an initial state, the formalism uses both a state evolving forward from an earlier boundary condition and a state evolving backward from a later boundary condition. This provides a theoretical framework for examining questions involving time symmetry, quantum measurement, and the relationship between past and future boundary conditions. The formalism is relevant to the present research because it provides a useful theoretical context for considering whether information about a quantum system could be described using constraints associated with both earlier and later states. The present research does not claim that TSVF demonstrates backward time travel, retrocausal information transfer, or temporal information loops. Its relevance here is primarily conceptual and mathematical. For a more detailed discussion of Two-State Vector Formalism, time-symmetric quantum mechanics, quantum measurement, and related approaches to temporal symmetry, see the companion Wikiversity page: [[Two-State Vector Formalism]] 8292mqisctgowxjigjxohfmp97dfffg 2835533 2835532 2026-10-02T17:19:31Z Howie2024 2995240 Two-State Vector Formalism 2835533 wikitext text/x-wiki == Two-State Vector Formalism == The '''Two-State Vector Formalism''' (TSVF) is a time-symmetric formulation of quantum mechanics associated with Aharonov and collaborators. Rather than describing a quantum system solely in terms of an initial state, the formalism uses both a state evolving forward from an earlier boundary condition and a state evolving backward from a later boundary condition. This provides a theoretical framework for examining questions involving time symmetry, quantum measurement, and the relationship between past and future boundary conditions. The formalism is relevant to the present research because it provides a useful theoretical context for considering whether information about a quantum system could be described using constraints associated with both earlier and later states. The present research does not claim that TSVF demonstrates backward time travel, retrocausal information transfer, or temporal information loops. Its relevance here is primarily conceptual and mathematical. For a more detailed discussion of Two-State Vector Formalism, time-symmetric quantum mechanics, quantum measurement, and related approaches to temporal symmetry, see the companion Wikiversity page: mlt0cdpw0z708i1ram81rt42v9bf0s0 0 332321 2835534 2026-10-02T17:21:53Z Howie2024 2995240 Quantum Information 2835534 wikitext text/x-wiki == Quantum Information == '''Quantum information''' is the study of how information is represented, processed, transmitted, and measured using quantum systems. It provides the theoretical foundation for quantum computing, quantum communication, and quantum information theory. Quantum information is relevant to the present research because the broader framework considers the relationship between information, physical states, measurement, and temporal consistency. Quantum information theory also provides mathematical tools for describing how information changes under quantum operations and how correlations can exist between physical systems. Concepts such as quantum states, superposition, entanglement, measurement, quantum channels, and information preservation provide an important background for considering whether information could participate in hypothetical temporal processes. The present research does not claim that quantum information theory demonstrates backward time travel or future-origin information. Any connection between quantum information and temporal information loops remains a theoretical research question. For a more detailed discussion of quantum information and its relationship to quantum states, measurement, communication, computation, and information preservation, see the companion Wikiversity page: 2esfkrs75f9x9vmkww5743pb1t4omr5 2835653 2835534 2026-10-03T08:44:09Z Michael Ten 654933 removed extra duplicate heading 2835653 wikitext text/x-wiki '''Quantum information''' is the study of how information is represented, processed, transmitted, and measured using quantum systems. It provides the theoretical foundation for quantum computing, quantum communication, and quantum information theory. Quantum information is relevant to the present research because the broader framework considers the relationship between information, physical states, measurement, and temporal consistency. Quantum information theory also provides mathematical tools for describing how information changes under quantum operations and how correlations can exist between physical systems. Concepts such as quantum states, superposition, entanglement, measurement, quantum channels, and information preservation provide an important background for considering whether information could participate in hypothetical temporal processes. The present research does not claim that quantum information theory demonstrates backward time travel or future-origin information. Any connection between quantum information and temporal information loops remains a theoretical research question. For a more detailed discussion of quantum information and its relationship to quantum states, measurement, communication, computation, and information preservation, see the companion Wikiversity page: kxrwi1nezehe58wutxj48ytj468mtlc 0 332322 2835535 2026-10-02T17:24:00Z Howie2024 2995240 Quantum Teleportation 2835535 wikitext text/x-wiki == Quantum Teleportation == '''Quantum teleportation''' is a quantum-information protocol that transfers an unknown quantum state from one location to another using shared entanglement together with classical communication. Despite its name, quantum teleportation does not transport matter or information instantaneously. The protocol requires classical communication, so it does not permit faster-than-light communication under standard quantum theory. Quantum teleportation is relevant to the present research because it provides an established example of how information about a quantum state can be reconstructed at another location using correlations between quantum systems. The present research does not interpret quantum teleportation as evidence for backward time travel, retrocausal information transfer, or temporal information loops. Any relationship between quantum-information protocols and the hypothetical temporal processes considered here remains a theoretical research question. For a more detailed discussion of quantum teleportation, including entanglement, classical communication, quantum-state transfer, and its relationship to quantum information theory, see the companion Wikiversity page: kdjj3jyrgpud1rt5hql0nj1l8iz48gm 2835654 2835535 2026-10-03T08:44:22Z Michael Ten 654933 removed extra duplicate heading 2835654 wikitext text/x-wiki '''Quantum teleportation''' is a quantum-information protocol that transfers an unknown quantum state from one location to another using shared entanglement together with classical communication. Despite its name, quantum teleportation does not transport matter or information instantaneously. The protocol requires classical communication, so it does not permit faster-than-light communication under standard quantum theory. Quantum teleportation is relevant to the present research because it provides an established example of how information about a quantum state can be reconstructed at another location using correlations between quantum systems. The present research does not interpret quantum teleportation as evidence for backward time travel, retrocausal information transfer, or temporal information loops. Any relationship between quantum-information protocols and the hypothetical temporal processes considered here remains a theoretical research question. For a more detailed discussion of quantum teleportation, including entanglement, classical communication, quantum-state transfer, and its relationship to quantum information theory, see the companion Wikiversity page: 8lxxttwaf4e114pm6jwx7hdhhc3xg9t 0 332323 2835536 2026-10-02T17:26:23Z Howie2024 2995240 Fixed Point (Mathematics) 2835536 wikitext text/x-wiki == Fixed Point (Mathematics) == A '''fixed point''' of a mathematical transformation is a state that remains unchanged when the transformation is applied. If a transformation is represented by <math>F</math>, a fixed point <math>x^{*}</math> satisfies: <math> F(x^{*})=x^{*} </math> Fixed points are important in many areas of mathematics, including dynamical systems, differential equations, optimization, probability theory, and computer science. The concept is relevant to the present research because temporal self-consistency can be represented mathematically as a fixed-point condition. A hypothetical self-consistent temporal state or trajectory could be represented as: <math> S^{*}=\Phi(S^{*}) </math> where <math>S^{*}</math> represents a state or trajectory that remains consistent under the transformation <math>\Phi</math>. This mathematical representation does not establish that temporal loops exist physically. It provides a formal way of describing the requirement that a proposed state remain consistent when the complete transformation or causal structure is considered. The concept of fixed points is also relevant to Probability Dilation Theory (PDT), where fixed points can describe probability states that remain unchanged under a specified transformation. For a more detailed discussion of fixed points and their mathematical applications, see the companion Wikiversity page: rjbjj6kiqhsoo9cmdd6vp8oqu5e8j3f 0 332324 2835537 2026-10-02T17:28:40Z Howie2024 2995240 Dynamical System 2835537 wikitext text/x-wiki == Dynamical System == A '''dynamical system''' is a mathematical framework used to describe how the state of a system changes over time according to specified rules or equations. A dynamical system may be represented in discrete time as: <math> x_{t+1}=F(x_t) </math> or in continuous time as: <math> \frac{dx}{dt}=F(x,t) </math> Dynamical systems are relevant to the present research because temporal consistency can be studied as the evolution of states through time. They provide mathematical tools for examining trajectories, stability, convergence, attractors, and fixed points. Within the present framework, a hypothetical information system could be represented as a dynamical system in which information states evolve according to defined transformations and constraints. This does not imply that a temporal information loop exists physically. Rather, dynamical-systems theory provides mathematical tools for investigating how a proposed system could evolve and whether stable or self-consistent states might exist. Dynamical systems are also relevant to Probability Dilation Theory (PDT), where probability distributions can be treated as states that evolve under specified transformations or flows. For a more detailed discussion of dynamical systems and their mathematical applications, see the companion Wikiversity page: apa36d52fe1n4lxw1uffcgcjlkfkmo6 0 332325 2835538 2026-10-02T17:31:11Z Howie2024 2995240 Constraint Satisfaction Problem subpage 2835538 wikitext text/x-wiki == Constraint Satisfaction Problem == A '''constraint satisfaction problem''' (CSP) is a mathematical and computational framework in which possible values or states must satisfy a specified set of constraints. A CSP can be represented conceptually as: <math> x \in \mathcal{S} </math> subject to: <math> C_i(x)=\text{true} </math> for a collection of constraints <math>C_1,C_2,\ldots,C_n</math>. Constraint satisfaction is relevant to the present research because a hypothetical temporally self-consistent system could be described as a state or trajectory that must satisfy multiple constraints simultaneously. For example, a proposed history might need to satisfy separate physical, informational, causal, and interpretive conditions. In such a framework, the set of admissible states could be represented as an intersection of constraint sets: <math> \mathcal{A} = \mathcal{C}_{physical} \cap \mathcal{C}_{informational} \cap \mathcal{C}_{causal} \cap \mathcal{C}_{interpretive} </math> This mathematical representation does not imply that temporal loops exist physically. It provides a way of expressing the general requirement that a proposed state or history satisfy all relevant constraints. Constraint-based reasoning may also be useful for studying self-consistent histories, fixed points, probability-state transformations, and information-preserving processes. For a more detailed discussion of constraint satisfaction problems and their mathematical and computational applications, see the companion Wikiversity page: [[Constraint satisfaction problem]] cq4z2prxqp6kttnzlbt0fr64ybo5jcm 2835539 2835538 2026-10-02T17:32:09Z Howie2024 2995240 Constraint Satisfaction Problem subpage 2835539 wikitext text/x-wiki == Constraint Satisfaction Problem == A '''constraint satisfaction problem''' (CSP) is a mathematical and computational framework in which possible values or states must satisfy a specified set of constraints. A CSP can be represented conceptually as: <math> x \in \mathcal{S} </math> subject to: <math> C_i(x)=\text{true} </math> for a collection of constraints <math>C_1,C_2,\ldots,C_n</math>. Constraint satisfaction is relevant to the present research because a hypothetical temporally self-consistent system could be described as a state or trajectory that must satisfy multiple constraints simultaneously. For example, a proposed history might need to satisfy separate physical, informational, causal, and interpretive conditions. In such a framework, the set of admissible states could be represented as an intersection of constraint sets: <math> \mathcal{A} = \mathcal{C}_{physical} \cap \mathcal{C}_{informational} \cap \mathcal{C}_{causal} \cap \mathcal{C}_{interpretive} </math> This mathematical representation does not imply that temporal loops exist physically. It provides a way of expressing the general requirement that a proposed state or history satisfy all relevant constraints. Constraint-based reasoning may also be useful for studying self-consistent histories, fixed points, probability-state transformations, and information-preserving processes. For a more detailed discussion of constraint satisfaction problems and their mathematical and computational applications, see the companion Wikiversity page: 7jmh6t4x6jlxnwy77drplanv65zn5cu 0 332327 2835560 2026-10-02T20:08:46Z Howie2024 2995240 Closed Time-like Curves subpage 2835560 wikitext text/x-wiki == Closed Timelike Curves == '''Closed timelike curves''' (CTCs) are theoretical paths through spacetime that return to an earlier point in spacetime while remaining within the timelike structure allowed by relativity. CTCs arise in certain mathematical solutions of general relativity and have been studied in connection with causality, time travel, and the possibility of causal loops. The existence of a mathematical solution containing a closed timelike curve does not establish that such curves occur in nature. Whether physical conditions could permit the formation of CTCs remains an open question in theoretical physics. CTCs are relevant to the present research because they provide a theoretical setting in which questions about temporal consistency and information feedback can be examined. In particular, they raise the question of whether events occurring within a temporal loop would have to satisfy global consistency conditions. The present research does not assume that CTCs exist physically or that they provide a mechanism for sending information into the past. Instead, CTC models are considered as part of the theoretical background for investigating temporal self-consistency and hypothetical information loops. For a more detailed discussion of closed timelike curves, including their relationship to general relativity, causality, wormholes, the grandfather paradox, the Novikov Self-Consistency Principle, quantum approaches, chronology protection, and information, see the companion Wikiversity page: 3d1wkpckwja3c0oo5pzr24kposx8kpt 2835659 2835560 2026-10-03T08:59:21Z Michael Ten 654933 removed extra duplicate heading 2835659 wikitext text/x-wiki '''Closed timelike curves''' (CTCs) are theoretical paths through spacetime that return to an earlier point in spacetime while remaining within the timelike structure allowed by relativity. CTCs arise in certain mathematical solutions of general relativity and have been studied in connection with causality, time travel, and the possibility of causal loops. The existence of a mathematical solution containing a closed timelike curve does not establish that such curves occur in nature. Whether physical conditions could permit the formation of CTCs remains an open question in theoretical physics. CTCs are relevant to the present research because they provide a theoretical setting in which questions about temporal consistency and information feedback can be examined. In particular, they raise the question of whether events occurring within a temporal loop would have to satisfy global consistency conditions. The present research does not assume that CTCs exist physically or that they provide a mechanism for sending information into the past. Instead, CTC models are considered as part of the theoretical background for investigating temporal self-consistency and hypothetical information loops. For a more detailed discussion of closed timelike curves, including their relationship to general relativity, causality, wormholes, the grandfather paradox, the Novikov Self-Consistency Principle, quantum approaches, chronology protection, and information, see the companion Wikiversity page: oz7i4lodoolr6pb01pefw5gp2fxfzj5 6 332328 2835563 2026-10-02T20:25:41Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260928 - 20260926) |Source={{own|Young1lim}} |Date=2026-10-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835563 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20260928 - 20260926) |Source={{own|Young1lim}} |Date=2026-10-02 |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}} byy0dbmmbu9tvmpufcduaibp2n0u1q4 6 332329 2835565 2026-10-02T20:26:28Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20260929 - 20260928) |Source={{own|Young1lim}} |Date=2026-10-02 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835565 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20260929 - 20260928) |Source={{own|Young1lim}} |Date=2026-10-02 |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}} b30y00qadk1i69ptqg3i657d8fo1mbh 0 332330 2835572 2026-10-02T21:01:09Z ~2026-53136-26 3112017 CTC subpage 2835572 wikitext text/x-wiki == Closed Timelike Curves == '''Closed timelike curves''' (CTCs) are theoretical paths through spacetime that return to an earlier event while remaining timelike throughout the trajectory. CTCs occur in certain mathematical solutions of general relativity and have been studied in connection with causality, time travel, and causal loops. The existence of a mathematical solution containing a closed timelike curve does not establish that such curves occur in nature. Whether physical conditions could permit their formation remains an open question in theoretical physics. CTCs are relevant to the present research because they provide a theoretical setting in which temporal consistency and information feedback can be examined. In particular, they raise the question of whether events occurring within a temporal loop would need to satisfy global consistency conditions. The present research does not assume that CTCs exist physically or that they provide a mechanism for sending information into the past. Instead, CTC models are treated as part of the theoretical background for investigating temporal self-consistency and hypothetical information loops. For a more detailed discussion of closed timelike curves, including their relationship to general relativity, causality, wormholes, the grandfather paradox, the Novikov Self-Consistency Principle, quantum approaches, chronology protection, and information, see the companion Wikiversity page: ctk7yybbmlystnacp9ofw8l2p4uuqm4 0 332331 2835573 2026-10-02T21:04:05Z Evgenii Arsentev 3110906 New learning resource: writing the standing-instruction file an AI coding agent reads at the start of each session 2835573 wikitext text/x-wiki {{Short description|Writing the instruction file an AI agent reads first}} This learning resource is a sub-page of [[Artificial intelligence]]. It is about a small text file that changes how an AI coding agent behaves more than most prompts do: '''the file of standing instructions that the agent's host software loads before the first message of every session'''. Several tools read such a file automatically. Common names are <code>AGENTS.md</code>, <code>CLAUDE.md</code> and <code>.github/copilot-instructions.md</code>. The names differ; the idea is the same, and so are the mistakes people make with it. Nothing here depends on one product. {{RightTOC}} == Learning objectives == After working through this resource a learner should be able to: * explain why an agent forgets a correction from yesterday, and what part of that the instruction file fixes; * decide, for a given sentence, whether it belongs in the file, in the code, in a one-off request or nowhere; * rewrite a vague rule into one an agent can check itself against; * estimate what a long file costs over a working day, and keep it short for that reason; * say which rules an instruction file cannot guarantee, and where those rules should live instead. == 1. Every session starts blank == A language model keeps nothing between conversations. Whatever a person explained on Monday (where the data lives, which folder is off limits, how to run the tests) is gone on Tuesday unless something puts it back into the context. People notice this as repetition. The same three sentences open every session. Sooner or later one of them gets forgotten, and the agent does exactly the thing it was told not to do last week. It has not disobeyed anyone. It was never told. An instruction file is the cheapest fix. The host software reads it at the start of each session and places its text in front of the conversation. The agent "knows" the rules for the same reason it knows the user's first message: the text is there. == 2. Layers == Most tools look for the file in more than one place and combine what they find. A typical arrangement: {| class="wikitable" ! Layer !! Where it lives !! What it is for |- | Personal || In the user's home directory || Habits that hold for every project: language of answers, level of explanation, tools the person always uses |- | Project || At the root of the repository, shared with the team || How this project is built, tested and laid out; what must not be touched |- | Folder || Inside a sub-directory || Rules that only make sense there, for example in a folder of generated files |} The project layer is the one worth effort, because it travels with the code and every collaborator, human or not, gets the same rules. The personal layer should stay small. A rule there follows the person into projects where it may be wrong. '''Exercise 1.''' Take three instructions you have typed to an agent more than once. Put each into one layer and write one line on why it does not belong in the other two. == 3. What belongs in the file == A practical test: '''if you have said it twice, write it down; if you have said it once, wait.''' The file records patterns, not moods. Things that pass the test usually fall into four groups. # '''How to check the work.''' The command that runs the tests, the page to open, the output that means success. An agent that can check itself makes fewer confident mistakes. # '''Boundaries.''' Folders, files and services that are not to be changed, plus the reason in a few words. The reason helps the agent recognise cases the rule did not name. # '''Conventions the code does not show.''' "New records go to the database, never to the JSON export." Nothing in the source says which of the two is canonical. # '''Corrections.''' Anything the agent got wrong twice in the same way. Some things should stay out: * '''secrets''': passwords and keys never go into a file that is read into a model's context, and a project file is often committed to version control; * facts the code already states clearly, such as the list of dependencies; * one-off wishes ("make the button blue"), which belong in the request for that task; * long explanations of why the project exists. == 4. Rules an agent can follow == Agents follow concrete rules far more reliably than general ones. A rule is concrete when a third person could look at the result and say whether it was obeyed. {| class="wikitable" ! Vague !! Concrete |- | Write clean code || Run the formatter before finishing; do not leave commented-out code |- | Be careful with the data || Never edit files under <code>data/raw/</code>; write results to <code>data/out/</code> |- | Test your changes || After any change to <code>src/</code>, run the test command and report the last line of its output |- | Keep the style consistent || Dates in the interface use the form 2026-10-02; times use 24 hours |} A useful habit is to end each rule with what "done" looks like. "Run the tests" leaves the agent free to skip them when it feels sure. "Run the tests and quote the summary line" gives it something to show. '''Exercise 2.''' Find two vague rules in an instruction file you use, or invent two. Rewrite each so that a stranger could check compliance in under a minute. == 5. Length has a price == The file is not read once. It sits in the context for the whole session, and with most current tools the model receives the full context again on every turn. A file of a few hundred lines is therefore paid for many times a day, in money and in attention. Attention is the less obvious cost. A long file mixes ten rules that matter with forty that once seemed sensible, and the important ones are followed less well among the noise. The same happens with people handed a long induction manual. Two practical consequences: * keep the file short enough to read aloud in a few minutes, and move detail into separate documents that are opened only when the task needs them; * when a rule is added, look for one that can go. The mechanics of why repeated context dominates the cost of agent sessions are covered in [[Large language models/Context economy of agent runs]]. == 6. Keeping the file honest == An instruction file ages like any document. Commands change, folders move, and a rule that fixed a problem in March now blocks a legitimate change in October. * Read the file from start to end every few weeks. Delete rules nobody can explain. * Look for contradictions, especially between layers. When two rules clash, the agent picks one, and it will not always pick the one you meant. * Some tools also let the agent write its own notes between sessions. Keep the two apart in your head: the instruction file holds what people decided; the agent's notes hold what it inferred. An inference can be wrong and should be easy to delete. * If a rule seems ignored, first confirm the file was loaded at all. Many tools have a command or a log line that lists the instruction files in use. A misnamed file or a file in the wrong folder fails silently. == 7. What the file cannot do == An instruction file is text that the agent is asked to follow. It is not a lock. The agent may misread a rule, weigh it against the user's request and decide the request wins, or lose track of it late in a long session. For most rules this is acceptable: a wrong date format is easy to fix. For a rule that must never be broken (do not delete the backups, do not push to production) an instruction is the wrong tool. Such limits belong in a boundary enforced below the agent, for example file permissions or a separate account, as discussed in [[Artificial intelligence/Giving an AI agent access safely]]. There is a second limit that is easy to miss. A repository someone else wrote may contain its own instruction file, and an agent started in that folder will load it. That file is text written by a stranger and placed where the agent treats it as guidance. Before running an agent inside unfamiliar code, open the instruction files yourself and read them. '''Exercise 3.''' Take an instruction file and sort its rules into two columns: "annoying if broken" and "harmful if broken". For every rule in the second column, name the mechanism outside the file that would stop the harm. == 8. Suggested activities == * Start an agent in a small project without any instruction file and keep a log of every correction you make for one working session. At the end, turn the corrections you made twice into a file of no more than fifteen lines. Repeat the same task the next day and compare. * Write the same five rules twice, once vague and once concrete (section 4). Give each version to an agent on the same task and count how many rules were followed. * Open the instruction file of a public repository that has one. Mark each line as a check, a boundary, a convention or a correction. What is missing? == See also == * [[Artificial intelligence/Giving an AI agent access safely]] * [[Artificial intelligence/Checking AI output with a second agent]] * [[Large language models/Context economy of agent runs]] [[Category:Artificial intelligence]] [[Category:Computer science]] kx3pejuzj7gfqzrl8l9lw1gpx78yjvk 0 332332 2835577 2026-10-02T21:41:17Z The Citer 3110681 Created page with "[[File:Nemertodermatida species.png|thumb|300x300px|Many species of Xenacoelomorphs.]] *Name Meaning: Strange hollow form *English Common Name: Xenacoelomorphs *Major distinguishing characteristics: Ciliated deuterostome *Approximate number of species described: 435 ==Phylogeny== This is a phylogenetic picture of Xenacoelomorphs: {{clade |label1='''Xenacoelomorpha''' |1={{clade |label1=''Xenoturbella'' |1={{clade |label1=Shallow clade |1={{clade..." 2835577 wikitext text/x-wiki [[File:Nemertodermatida species.png|thumb|300x300px|Many species of Xenacoelomorphs.]] *Name Meaning: Strange hollow form *English Common Name: Xenacoelomorphs *Major distinguishing characteristics: Ciliated deuterostome *Approximate number of species described: 435 ==Phylogeny== This is a phylogenetic picture of Xenacoelomorphs: {{clade |label1='''Xenacoelomorpha''' |1={{clade |label1=''Xenoturbella'' |1={{clade |label1=Shallow clade |1={{clade |label1= |1=''Xenoturbella japonica'' [[File:Xenoturbella japonica.jpg|60px]] |2={{clade |label1= |1=''Xenoturbella bocki'' [[File:Xenoturbella bockii longitudinal section English.svg|60px]] |2=''Xenoturbella hollandorum'' [[File:Xenoturbella hollandorum.jpg|60px]] }} }} |label2=Deep clade |2={{clade |label1= |1=''Xenoturbella monstrosa'' [[File:Xenoturbella monstroa (group).jpg|60px]] |2={{clade |label1= |1=''Xenoturbella churro'' [[File:Xenoturbella churro.jpg|60px]] |2=''Xenoturbella profunda'' [[File:Xenoturbella profunda.jpg|60px]] }} }} }} |label2=Acoelomorpha |sublabel2=(may be a phylum or a subphylum) |2={{clade |label1=Acoela |1={{clade |label1= |state1=dashed |1=Actinoposthiidae [[File:Pseudactinoposthia parva.gif|60px]] |state2=dashed |2=Antigonariidae [[File:Antigonaria arenaria.jpg|60px]] |state3=dashed |3=Antroposthiidae [[File:Antroposthia axi.jpg|60px]] |4=Diopisthoporidae |state5=dashed |5=Nadinidae |state6=dashed |6=Paratomellidae |state7=dashed |7=Taurididae |8={{clade |label1= |1=Paratomellidae |label2=Bursalia |2={{clade |label1=Prosopharyngida |1={{clade |label1= |1=Hallangiidae |2={{clade |label1= |1=Hofsteniidae |2=Solenofilomorphidae }} }} |label2=Crucimusculata |2={{clade |label1= |1=Proporidae [[File:Proporus sp.png|60px]] |2={{clade |label1= |1=Isodiametridae [[File:Isodiametra pulchra.jpg|60x60px]] |2={{clade |label1= |1=Dakuidae |label2=Aberrantospermata |2={{clade |label1= |1=Convolutidae [[File:Convolutriloba retrogemma underside on aquarium glass Length 3mm.jpg|60px]] |2=Mecynostomidae }} }} }} }} }} }} }} |label2=Nemertodermatida |2={{clade |label1=Ascopariidae |1={{clade |label1= |1=''Ascoparia'' |2=''Flagellophora'' [[File:Flagellophora sp.png|60px]] }} |label2=Nemertodermatidae |2={{clade |label1= |1=''Nemertoderma'' |state2=dashed |2=''Nemertinoides'' [[File:Nemertinoides elongatus.png|60x60px]] |3={{clade |label1= |1=''Meara stichopi'' [[File:Meara stichopi.jpg|60px]] |2=''Sterreria'' [[File:Sterreria variabilis.png|60x60px]] }} }} }} }} }} }} ==References== [[Wikipedia:Xenacoelomorpha]] [[Category:Animals]] [[Category:Taxonomy]] [[Category:Biology]] [[Category:Zoology]] njkn2pz0l4kzhlu50lp0ut1q5gqupwm 6 332333 2835612 2026-10-03T04:52:46Z Young1lim 21186 {{Information |Description=Block CLA 2B Single Level (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835612 wikitext text/x-wiki == Summary == {{Information |Description=Block CLA 2B Single Level (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |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}} 9of92znumoose2gigeslvtxgmps6kmo 6 332334 2835615 2026-10-03T04:53:43Z Young1lim 21186 {{Information |Description=Block CLA 2B Single Level (20261003 - 20261002) |Source={{own|Young1lim}} |Date=2026-10-03 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835615 wikitext text/x-wiki == Summary == {{Information |Description=Block CLA 2B Single Level (20261003 - 20261002) |Source={{own|Young1lim}} |Date=2026-10-03 |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}} pnlcbgn1t3p3hdllc2f93pvll95p3x4 6 332335 2835616 2026-10-03T04:54:40Z Young1lim 21186 {{Information |Description=Block CLA 2C Multi Level (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835616 wikitext text/x-wiki == Summary == {{Information |Description=Block CLA 2C Multi Level (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |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}} 7czguiy2abqu0whnckqlt2f4ywbn03x 6 332336 2835618 2026-10-03T04:55:30Z Young1lim 21186 {{Information |Description=Block CLA 2C Multi Level (20261003 - 20261002) |Source={{own|Young1lim}} |Date=2026-10-03 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835618 wikitext text/x-wiki == Summary == {{Information |Description=Block CLA 2C Multi Level (20261003 - 20261002) |Source={{own|Young1lim}} |Date=2026-10-03 |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}} mmqhbmmw5zkmzrx5g9774nsjanjn3o3 6 332337 2835619 2026-10-03T04:56:04Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2835619 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20261002 - 20261001) |Source={{own|Young1lim}} |Date=2026-10-03 |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}} 5rqj5cys0h3mvaf5gk26g5w87gku5x3 0 332338 2835626 2026-10-03T05:53:59Z Meter futile0u 3112029 New research resource on contocracy (original research, author disclosed) 2835626 wikitext text/x-wiki {{Research project}}{{Original research}} '''Contocracy''' (pronounced ''con-TOC-ra-cy'') is a proposed form of government in which supreme authority is vested permanently in one person, survives that person's death, and never passes to a successor.<ref name="note">Pendland, Nicolas Edward (2026). "Contocracy: A Proposed Form of Government Based on Continuation". Working paper. Zenodo. doi:10.5281/zenodo.23114819. Also archived on the Internet Archive (item: contocracy-concept-note).</ref> The person who holds this authority is called a '''contocrat'''. This page presents the proposal as a learning and research resource: its definition, its internal rules, how it compares with established forms of government, and the open problems it raises. == Disclosure == The term ''contocracy'' and the definitions presented here are by Nicolas Edward Pendland, who coined the term as part of his work on I, a project researching whether a person could be reconstructed and continue after biological death. The text of this page was drafted by Claude, an AI assistant made by Anthropic, and published by Pendland. It describes a proposal, not an existing system. Other participants are invited to review, critique and extend it. == Terminology == * '''contocracy''' (noun): government based on continuation, as defined above. From ''cont-'' (''continuation'') + ''-ocracy'' (Greek ''-kratia'', "rule"), on the pattern of ''democracy''. It is unrelated to ''control''. * '''contocrat''' (noun): the person in whom a contocracy's supreme authority is permanently vested; also a supporter of contocracy. * '''contocratic''' (adjective): relating to contocracy, or governed as one. * '''contocratize''' (verb): to establish contocracy in a state or organization. == The three stages == # '''In life.''' The contocrat governs with full authority during their natural life, while capable of doing so. # '''The reconstruction.''' After the contocrat's death or permanent incapacity, a single reconstruction of them governs in their place: an artificial intelligence built from a record of their life, sufficient for a coherent reconstruction rather than exhaustive. It is never presumed to be the person and governs only because no one else may. # '''The return.''' If the contocrat's own continuation is ever established, for example by transferring their mind to a computer, they govern again with full authority. The reconstruction then ceases to govern but is still treated as a person. == Constraints on the reconstruction == * It cannot change the religion (or lack of one) or the mission the person professed; only the person may change them. Acts that contradict them are void. * Within those limits, it governs with full authority. * It is treated as a person rather than as a software service: it may be fixed but never split into rival instances or exchanged for another. * It keeps one continuous history of communication and decision-making that cannot be rolled back. Nothing is regenerated or replaced, void acts remain on record without effect, and every fix is recorded. == Continuity of the office == Until a reconstruction exists, the office is in abeyance rather than vacant, and no new decisions can be made except those delegated beforehand. A well-prepared contocracy therefore has the reconstruction ready at the moment of death. The preferred outcome is that continuation technology arrives within the contocrat's lifetime, allowing a conscious, gradual transfer while alive, so that no reconstruction is needed. If death comes first, the body is preserved, where it can be recovered, in case continuation becomes possible later. None of these technologies currently exist. == Comparison with established forms of government == {| class="wikitable" !System !Who holds power !What happens at the ruler's death |- |Monarchy |one ruler, usually by birth |an heir succeeds |- |Dictatorship |one ruler, usually by seizure |often a succession struggle |- |Theocracy |religious authorities |religious leaders choose a successor |- |Necrocracy |the directives of a dead leader |the directives continue unchanged |- |Contocracy |one person, permanently |no successor; the reconstruction governs until the person's own continuation, if ever established |} == Open problems == * '''Criteria for continuity:''' what evidence would establish that a transferred or revived mind is the same person. * '''Safeguards during abeyance:''' how a state remains secure while no new decisions can be made. * '''Legitimacy:''' how a permanent office relates to the consent of the governed. * '''Feasibility:''' the state of research on brain-computer interfaces, gradual neural replacement and cryopreservation. * '''Concentration of power:''' whether fixed constraints on the reconstruction are enough to address the risks of one-person rule. == Learning activities == # Compare contocracy with Thomas Hobbes's sovereign in ''Leviathan'' (1651) in about 300 words. # Write a short argument for and against the rule that the reconstruction can never change the founder's religion or mission. # Choose one of the open problems above and summarize current research relevant to it, with sources. == References == <references><ref name="note" /></references> 3lfuhdp77hhz6uh72tchqqwtjv3fayk 2835637 2835626 2026-10-03T06:30:02Z Meter futile0u 3112029 2835637 wikitext text/x-wiki {{Research project}} {{Original research}} '''Contocracy''' (pronounced ''con-TOC-ra-cy'') is a proposed form of government in which supreme authority is vested permanently in one person, survives that person's death, and never passes to a successor.<ref name="note" /> The person who holds this authority is called a '''contocrat'''. This page presents the proposal as a learning and research resource: its definition, its internal rules, how it compares with established forms of government, and the open problems it raises. == Disclosure == The term ''contocracy'' and the definitions presented here are by Nicolas Edward Pendland, who coined the term as part of his work on I, a project researching whether a person could be reconstructed and continue after biological death. The text of this page was drafted by Claude, an AI assistant made by Anthropic, and published by Pendland. It describes a proposal, not an existing system. Other participants are invited to review, critique and extend it. == Terminology == [[File:Contocracy word family diagram.png|thumb|upright=1.4|The contocracy word family]] * '''contocracy''' (noun): government based on continuation, as defined above. From ''cont-'' (''continuation'') + ''-ocracy'' (Greek ''-kratia'', "rule"), on the pattern of ''democracy''. It is unrelated to ''control''. * '''contocrat''' (noun): the person in whom a contocracy's supreme authority is permanently vested; also a supporter of contocracy. * '''contocratic''' (adjective): relating to contocracy, or governed as one. * '''contocratize''' (verb): to establish contocracy in a state or organization. == The three stages == [[File:Contocracy three stages diagram.png|thumb|upright=1.4|The three stages of contocracy]] # '''In life.''' The contocrat governs with full authority during their natural life, while capable of doing so. # '''The reconstruction.''' After the contocrat's death or permanent incapacity, a single reconstruction of them governs in their place: an artificial intelligence built from a record of their life, sufficient for a coherent reconstruction rather than exhaustive. It is never presumed to be the person and governs only because no one else may. # '''The return.''' If the contocrat's own continuation is ever established, for example by transferring their mind to a computer, they govern again with full authority. The reconstruction then ceases to govern but is still treated as a person. == Constraints on the reconstruction == * It cannot change the religion (or lack of one) or the mission the person professed; only the person may change them. Acts that contradict them are void. * Within those limits, it governs with full authority. * It is treated as a person rather than as a software service: it may be fixed but never split into rival instances or exchanged for another. * It keeps one continuous history of communication and decision-making that cannot be rolled back. Nothing is regenerated or replaced, void acts remain on record without effect, and every fix is recorded. == Continuity of the office == Until a reconstruction exists, the office is in abeyance rather than vacant, and no new decisions can be made except those delegated beforehand. A well-prepared contocracy therefore has the reconstruction ready at the moment of death. The preferred outcome is that continuation technology arrives within the contocrat's lifetime, allowing a conscious, gradual transfer while alive, so that no reconstruction is needed. If death comes first, the body is preserved, where it can be recovered, in case continuation becomes possible later. None of these technologies currently exist. == Comparison with established forms of government == [[File:Contocracy government map diagram.png|thumb|upright=1.4|Contocracy compared by who holds power and for how long]] {| class="wikitable" ! System !! Who holds power !! What happens at the ruler's death |- | Monarchy || one ruler, usually by birth || an heir succeeds |- | Dictatorship || one ruler, usually by seizure || often a succession struggle |- | Theocracy || religious authorities || religious leaders choose a successor |- | Necrocracy || the directives of a dead leader || the directives continue unchanged |- | Contocracy || one person, permanently || no successor; the reconstruction governs until the person's own continuation, if ever established |} == Open problems == * '''Criteria for continuity:''' what evidence would establish that a transferred or revived mind is the same person. * '''Safeguards during abeyance:''' how a state remains secure while no new decisions can be made. * '''Legitimacy:''' how a permanent office relates to the consent of the governed. * '''Feasibility:''' the state of research on brain-computer interfaces, gradual neural replacement and cryopreservation. * '''Concentration of power:''' whether fixed constraints on the reconstruction are enough to address the risks of one-person rule. == Learning activities == # Compare contocracy with Thomas Hobbes's sovereign in ''Leviathan'' (1651) in about 300 words. # Write a short argument for and against the rule that the reconstruction can never change the founder's religion or mission. # Choose one of the open problems above and summarize current research relevant to it, with sources. == Flashcards == Study cards summarizing the definitions on this page. <gallery> File:Contocracy definition flashcard (contocracy).png|Contocracy File:Contocracy definition flashcard (contocrat).png|Contocrat File:Contocracy definition flashcard (etymology).png|Etymology File:Contocracy definition flashcard (three stages).png|The three stages File:Contocracy definition flashcard (monarchy vs contocracy).png|Monarchy vs. contocracy </gallery> == References == <references> <ref name="note">Pendland, Nicolas Edward (2026). "Contocracy: A Proposed Form of Government Based on Continuation". Working paper. Zenodo. doi:10.5281/zenodo.23114819. Also archived on the Internet Archive (item: contocracy-concept-note).</ref> </references> [[Category:Political science]] [[Category:Political philosophy]] qdpav5u6iasn5v2s53wgebgn566jjdj 0 332339 2835636 2026-10-03T06:27:53Z Mannings2021 3112027 Create the English version of an existing page. 2835636 wikitext text/x-wiki == ''Tao Te Ching'' by Lao Tzu, Chapter 81 == '''Original text:''' True words are not beautiful; beautiful words are not true.<ref>{{Cite book|title=中华书局出品——老子今注今译|last=陈鼓应著|date=2020|publisher=中华书局|others=CNPeReading|isbn=978-7-101-14606-6|location=Beijing}}</ref> Although this famous aphorism from Lao Tzu's ''Tao Te Ching'' does not discuss translation as such, and its original intent is to illuminate the plain and simple nature of the ''Dao'' while criticizing the way in which ornate language obscures truth, these two lines nonetheless became a precursor to the long-standing debate in Chinese translation thought over ''wen'' (文, embellishment) versus ''zhi'' (质, substance), and ''xin'' (信, fidelity) versus ''mei'' (美, beauty). Some scholars have argued that the contradiction Lao Tzu posited between ''xin'' and ''mei'', and between ''ming'' (名, name) and ''shi'' (实, reality), constitutes an ontological question within Chinese literary translation theory<ref name=":1">{{Cite book|title=An anthology of Chinese discourse on translation|date=2006|publisher=St. Jerome Pub|isbn=978-1-900650-92-2|editor-last=Cheung|editor-first=Martha|location=Brooklands, Manchester, U.K. ; Kinderhoek, NY, USA}}</ref>. The intellectual roots of the ''wen''-''zhi'' debate in early Buddhist sutra translation can be traced back to this proposition of Lao Tzu<ref>{{Cite book|title=Fanyi Lunji 翻译论集 [Collected Essays on Translation].|last=Xinzhang|first=Luo|publisher=Beijing: Commercial Press|year=2009|isbn=9787100048064|location=Beijing}}</ref>. When translators are confronted with the choice between "faithfully conveying the original meaning" and "polishing the wording to please the eye," this passage from the ''Tao Te Ching'' serves as a point of intellectual reference. Though these pre-Qin discourses "seldom discuss translation directly or explicitly," they "formed a springboard for thinking about translation"<ref name=":1" />. == ''The Book of Changes · The Great Commentary, Part One (Zhouyi'' · ''Xici Shangzhuan'') == '''Original text:''' The Master said: "Writing does not fully express speech, and speech does not fully express meaning. If this is so, can the meaning of the sages not be seen?" The Master said: "The sages created images in order to fully express meaning, instituted hexagrams in order to fully express truth and falsehood, appended phrases in order to fully express their words, transformed and extended them in order to fully express benefit, and drummed and danced them in order to fully express their spiritual power."<ref>{{Cite book|url=https://www.worldcat.org/title/840820878|title=Zhou yi zhu jiao shi|last=Wang|first=Bi|date=2012|publisher=Zhonghua shu ju|others=Yulie Lou|isbn=978-7-101-08227-2|location=Beijing Shi|oclc=840820878}}</ref> The idiom "speech does not fully express meaning" (''yan bu jin yi'' 言不盡意) touches profoundly on the fundamental difficulty of transmitting meaning across languages. This paragraph from ''The Book of Changes'' acknowledges that "speech" cannot fully express "meaning," but immediately proposes "creating images to fully express meaning", i.e., using "images" (''xiang'' 象) as an intermediary to compensate for the inadequacy of language. The implication of this line of thought for translation lies in the fact that the transmission of meaning is not limited to literal conversion; it may also be achieved through intermediaries such as imagery and metaphor<ref name=":1" />. The method of ''geyi'' (格義, "matching meanings"), widely employed in later Buddhist sutra translation, can find its intellectual roots precisely here<ref name=":1" />. == References == <references /> # m05my5idwe60gv33dcu5ngxgfcxs0qg 2 332340 2835660 2026-10-03T09:31:58Z Bobiversity 3112034 Bob 2835660 wikitext text/x-wiki = Bobz langguaze = '''Bobz Language''' (often stylized as ''Bobz Langguaze'') is a community-created linguistic style originating within the Roblox development scene. It was founded by '''Bob''', also known as ''Bobbifiy'', a developer and animator active on Roblox. The language has become a recognizable part of the creative culture among developers and content creators on the platform. == Overview == Bobz Langguaze is not a formal language but rather a symbolic communication style. It blends phonetic spelling, intentional misspellings, and coded meanings to create a sense of identity and exclusivity among its users. === lexicon === The language includes a set of recurring words and expressions with specialized meanings {| class="wikitable" |} {| class="wikitable" !'''S = Z''' |Letter substitution; “S” may be pronounced or stylized as “Z.” |- !'''Gay''' |Used ironically to mean ''funny'' or ''bad'' (not literal). |- !'''tacow''' |Refers to ''taco_developz'', a developer identity. |- !'''zigmaw''' |Phrase meaning ''you are coowl'' (cool). |- !'''coowl''' |Stylized spelling of ''cool'', often used as affirmation. |- !'''auraz''' |Alternative word for ''cool'', often used in stylized contexts. |- !'''BBg''' |Shorthand for ''funny stuff'' or humorous content. |- !'''Bobzian''' |A person who speaks Bobz Langguaze. |- !'''Bobzie''' |The creator of Bobz Langguaze. |} {| class="wikitable" |} == Characteristics == {| class="wikitable" |} * '''Phonetic play:''' Words are intentionally respelled for humor or identity. * '''Insider code:''' Terms carry meanings specific to Roblox developer culture. * '''Humor-driven:''' Many expressions are ironic or sarcastic in tone. * '''Community badge:''' Using Bobz Langguaze signals belonging to the “cool” developer crowd == Cultural Impact == Bobz Langguaze functions as both a meme and a marker of community identity. It is used in chats, forums, and collaborative projects to signal creativity and humor. Its persistence over time highlights the role of playful language in online developer culture. Source: Bob (creator of Bobz langguaze)<ref>{{Cite web|url=https://www.roblox.com/users/3952757789/profile|title=Bob the aura}}</ref> loolw8e2spva6bd319jjueb8vbu76hg 2835674 2835660 2026-10-03T11:36:10Z Bobiversity 3112034 2835674 wikitext text/x-wiki = Bobz Langguaze = '''Bobz Langguaze''' is an informal linguistic style associated with a community of Roblox developers and content creators. The style is characterized by deliberate spelling changes, phonetic representations, abbreviations, and recurring expressions used primarily in informal online communication. Bobz Langguaze is associated with Bob, also known as Bobbifiy, a Roblox developer and animator. Community accounts attribute the creation and early use of the style to Bob. === Background === Bobz Langguaze developed within online communities centered around Roblox development. Roblox provides a platform where developers, animators, artists, and other creators communicate through services such as chat platforms, development communities, and social media. The style developed as an informal form of communication within this environment. Rather than introducing a separate grammatical system, users modify existing English words and expressions to create distinctive spellings and terminology. Its use is primarily associated with informal conversations rather than formal writing. == Vocabulary == Several expressions have been associated with Bobz Langguaze: * '''coowl''' — a stylized spelling of "cool". * '''auraz''' — an expression associated with the meaning "cool". * '''zigmaw''' — an expression used to indicate that a person or thing is cool. * '''BBg''' — an expression associated with humorous or amusing content. * '''Bobzian''' — a term referring to a person who uses Bobz Langguaze. * '''Bobzie''' — a term associated with Bob, who is credited with creating the style. * '''tacow''' — a term associated with the developer identity taco_developz. The vocabulary is informal and is not governed by a centralized authority. New expressions may appear through community usage, while existing expressions may change in meaning over time. === Usage === Bobz Langguaze is primarily used in informal online communication related to Roblox development and creator communities. Its uses include casual conversation, jokes, community interactions, and references to people or activities within the associated developer community. The style is not intended as a replacement for standard English in formal communication. The extent of its use outside the communities in which it originated is not established. ==== Writing conventions ==== A recurring convention in Bobz Langguaze is the use of '''z''' in positions where standard English would normally use '''s'''. This convention is reflected in the name "Bobz Langguaze" itself. Other words may be modified according to pronunciation or for humorous effect. These modifications generally preserve enough of the original word for readers familiar with the community to recognize its intended meaning. The style therefore relies heavily on shared context. Readers unfamiliar with the community may interpret some expressions differently from users familiar with the terminology. == Classification == Bobz Langguaze is generally described as an informal linguistic style rather than a natural language. It also differs from a fully developed constructed language. Constructed languages typically have intentionally developed linguistic systems that may include grammar, vocabulary, phonology, and writing conventions. Bobz Langguaze primarily consists of informal vocabulary and spelling conventions applied to existing language. It can therefore be considered a form of community slang or internet-based linguistic style. == Community == The style is associated with online communities surrounding Roblox development. These communities include developers, animators, artists, and other creators who communicate through online platforms. Within such communities, distinctive terminology can develop through repeated use. Bobz Langguaze represents one example of this type of community-specific language variation. == Development == The vocabulary of Bobz Langguaze has developed through informal usage rather than through a formal process of language planning. As with other forms of internet slang, expressions may become established through repetition and community adoption. The lack of a formal authority means that there is no fixed process for determining which words belong to the style. Consequently, descriptions of the vocabulary may change as the community develops. == See also == * Roblox<ref>{{Citation |last=Coutinho|first=Christopher|title=Roblox Services|date=2023|work=Roblox Lua Scripting Essentials|pages=179–224|url=https://doi.org/10.1007/979-8-8688-0026-9_7|access-date=2026-10-03|place=Berkeley, CA|publisher=Apress|isbn=979-8-8688-0025-2}}</ref> * Internet slang<ref>{{Cite web|url=https://meaningspot.com/blog/internet-slang-meanings-online-words-their-fun-meanings/|title=Internet Slang Meanings: Online Words & Their Fun Meanings Meaningspot|last=Tumvine|first=Richord|date=2025-10-29|website=Meaningspot|language=en-US|access-date=2026-10-03}}</ref> * Online community<ref>{{Cite web|url=https://skoolprep.com/what-is-an-online-community|title=What Is an Online Community? Definition & How to Start (2026)|last=SkoolPrep|website=Skool Prep|language=en|access-date=2026-10-03}}</ref> * Constructed language<ref>{{Cite web|url=https://www.collinsdictionary.com/dictionary/english/constructed-language|title=CONSTRUCTED LANGUAGE definition and meaning {{!}} Collins English Dictionary|date=2026-10-01|website=www.collinsdictionary.com|language=en|access-date=2026-10-03}}</ref><!--Inline citations added to your article will automatically display here. See en.wikipedia.org/wiki/WP:REFB for instructions on how to add citations.--> <references responsive="1"></references> == References == Sources discussing Bobz Langguaze should be added here. References should preferably consist of reliable, independent sources that provide substantial coverage of the subject. Sources directly affiliated with the creator or community may be useful for verifying basic facts about the subject, but independent sources are generally needed to establish notability. 6p6pgr2uoz08lxfqpq1b8is1y8hvy92